Choddo,
Yeah, the "cover with tongue" was cracked, so you're seeing the cable in there. The stuff that is rubbing off was actually above that area, in the housing above the barrel adjuster. I should have my bike back tomorrow, and I'm going to remove that housing and inspect the cable. I am also going to cut a longer housing for that part, because I can tell that the housing piece that the bike shop cut is too short. The housing is not quite kinked but almost, and it's bowing, which means there is pressure there and there should not be. That can have a huge impact on shifting. So I'll sort that out tomorrow, see if it helps.

I’ve seen quite a few rear mechs where the cable comes out of there at quite an angle, and bends that sleeve, in certain gears so it might not necessarily be a problem and I’m not sure there’s much you can do, the cable is taut there but yeah the last run of housing on the other side should not kink.

If the plan is still to take it back to the shop that did the cabling install, probably should take a picture of the cabling at the RD so the 'owner' can see the work that was done.

If the 'coating' is damaged in the area that is within the outer housing, then the install was not done too well. Yes, a too short outer causes the inner cable to bend and 'snag' on the housing, or the end was poorly cut, causing drag on the inner.

Hi KCT,
Yes, I still need to take it back to the shop that did the cabling, because there is also an issue with the front derailleur. The L limit screw on the front is screwed all the way down, and it wasn't like that before and it shouldn't be that way. It makes it impossible to fully/properly adjust the FD, which is also not shifting all that great still.
I fully intend to take pics/video of the RD housing before I mess with putting on a longer housing/check the cable. That cable wear was inside the housing around/above the barrel adjuster, and I know that is wrong. I hate to be this way, but if a longer housing fixes or helps the shifting issue, I'm going to put the shorter one back on there and keep the one I cut, to show them their mistake when I go to their shop next Tuesday. I want to know two things from the shop manager: if he thinks the cable is too short and will affect the shifting, or if he thinks it's fine and blows it off as nothing. I don't trust this shop at all at this point, and I want to prove a point. I emailed the shop manager pics of the same thing I showed you all, the broken cover/fraying cable, and he literally said it was not a cause for concern. That bothered me. How in the world is a brand new cable fraying and a broken piece not a concern??
Anyhoo. I'll update once I get this housing checked, which should be this afternoon.

Update.
I got the bike home, with a few longer pieces of housing that I cut at the LBS. I was about to remove the cable from the housing when I noticed that the end of the cable was slightly frayed. I grabbed a new ferrule, and tried to push it through, and it will not go because it was frayed a little tiny bit. It's even worse now, because I had to mess with it a lot to get all the slack out of the cable when I reattached it to the pinch bolt. Is my new cable screwed now?

That housing looks fine to me.
You could have a peek under part of the housing by letting the cable back out as far as the pinch bolt. Do not cut it where you pointed because you won’t have enough to grab to tension it when you re-tighten it. You can do it by just having it loose and moving the mech into position I guess but probably needs another pair of hands.

It’s a mess but put a new endcap on the cable now before it gets any worse.

any chance you can do a video of it shifting up and down the cassette in the stand there?

As far as the tension, your video shows that you are in the next to highest gear. I would expect tension being there otherwise the derailleur would not have been able to shift down. I agree with [MENTION=498230]choddo[/MENTION] that the housing looks fine.

If you really think the bend is to sharp, buy a new cable, install the longer section of housing and invest in a few end crimps.

Thanks for the input you two. I did buy a new cable today, and I'm going to learn to put it on! This way I can change the housing and see if it's any better. JoeTBM, I shifted it down to the lowest gear and it still has the same tension. Will see what happens.
I made a video of it shifting in the stand as requested.

So this started before you did all the work?

I think people looking for a video desired to see a close up of the rear derailleur, not a wide shot of your bike featuring the shift lever.

Yes, this started before I did all the work. Why else would I be doing all of this? :)

Good point. I'll do a better one today!

Alright, this is the best I can do, video wise.

Hmm. That isn’t perfect, bit of hesitation on that 6th cog downshift but personally I’d be happy with that. FD could maybe come a bit closer to the frame as you say to avoid the rubbing there but that big:1st/2nd cog aren’t needed and you might introduce a bigger problem.

Yeah, the bike shop that did the cabling messed up the front too. The L limit screw on the FD is all the way in, and it should not be. It's impossible to properly adjust the FD at this point due to that. I'm not going to worry about that right now, as I'm going to make the shop fix that next Tuesday (as well as the rear shifting if I can't figure it out by then!).

Here is something else that I've noticed since I got the new chain, and I think it's an issue that could be contributing to this. I have a new chain, new cassette, and you'd think it would be silky smooth. But notice when I backpedal, the chain seems to hang up a bit. I have literally gone through the entire chain and checked each link for movement, to make sure none of them were sticking. I checked the lateral movement at each link as well when I did that. I made sure it's lubed properly as well. Here is a video to show what I mean.

You using a Shimano chain?

Yes. I asked the LBS for the CN-701 Ultegra chain, as that was the original chain, but after the chain was replaced I did some checking and it's the 601 (105) version. I wasn't too happy about that at all, so I bought a 701 from Jenson last night, it will be here next week. Weird thing about it, when I went to a second bike shop to try to get the shifting fixed again, they told me the chain was on backwards, and they said they fixed it. I took the chain off myself yesterday and went through it, checking for stiff links, etc. and it seems fine until I get it on the bike and pedal.

STOP!

Ever hear the saying that a camel is a horse designed by a committee?

IMO that's what's happening here. Too many people chiming in based on limited info, confusing the issue, and causing you to waste money.

Take the time to completely analyze the problem, and come up with a theory before making changes or spending another dime.

FWIW, I doubt it's the chain, or length of rear loop. As for your earlier reference to tension in that rear loop, that's not anything, since cable tension/housing compression are always equal, and determined by the RD's return spring.

For my money, I'd focus on cable friction, or something like a poorly cut end, a poorly seated ferrule, or something similar. Or poor action within the shifter.

But, don't go with my theory, because not having seen it, I can only guess along with everyone else here.

Seriously, find someone who knows his compost, maybe through a solid referral. Give him (her?) the bike with no info except "shifting is off" and let him work his magic unassisted. Answer questions, but don't offer any theories.

At this point, I suspect no parts are needed, except possible replacement of a butchered cable. However, if he suggests replacing the chain or whatever else has already been tried, let him know that.

Pretty sure no one has told him to spend anything on anything yet. He keeps doing that himself.

bikenoobDomane3 that clicking is weird but a 601 chain would be, imho, indistinguishable from a 701. The Shimano chains have arrows stamped on them (edit: just looked at a youtube video to double check this and it said the little text stampings should face you - I can’t see them in your video?) to check if they’re installed the right way. Does it do that in all the sprockets or just the smallest? If you look at it from behind is it riding up on to the next cog for a moment when it does that?

The FD L limit screw being all the way in does not affect that rubbing in big/small. As long as it drops onto the small ring ok that doesn’t matter.

FBinNY,
I hear ya, and no one has told me to spend money and such. I bought a new cable because I butchered the other one (live and learn) and I bought the Ultegra chain because that is what I wanted in the first place. I already had a Giant Defy 1 with 105 stuff, I upgraded to this Domane w/Ultegra for a reason, and dog gone it I want all Ultegra replacement parts haha.
That being said, hey, guess what? I did analyze and think about things these last few weeks since this started (3 weeks to be exact, driving me crazy the whole time) and I boiled it down to what you did, the cabling/friction. I installed the new cable this afternoon, and guess what? I think I fixed it!!! When I took out the old cable, there was more friction than I thought there should be, and I was feeling the cable as I went along. There were multiple spots on the cable that were getting worn already, and I think that the bike shop somehow got the RD cable mixed up with the FD cable. I threaded in the new cable, never done that before (took your advice to learn!), and already, without any adjustment yet, it's better than it was! I also made the housing at the RD longer. There was a place on the frame where the cable goes from the downtube to the chainstay, and there should have been a donut there, and it was missing, causing wear right at that point for sure. Which is interesting, because when I got my bike home from that bike shop, I found a donut on the floor a bit after. They didn't do it right, twice! The noob here did it right the first time. I may not be a bike mechanic yet, but I was a mechanic in the Army and I worked IT for 15 years, so I know how to problem solve and fix compost (hehe).
Anyways, I have to fix the FD now, because they messed that up too. Once that is done and I've got it all adjusted, I'll upload a video of how it should shift (if I did this right, that is. I might be celebrating a bit early, but I hope not, all signs point to not).
So, off to continue the fix!

Happy to hear it's coming together. As you've no doubt figured out (or already knew) the secret is to think faster than you work.

Use the clues the bike offers to list and rank possibilities and form a theory before jumping in.

It seems part of the issue was being ill served by sloppy professional(?) shop mechanics, and hopefully you won't need their help going forward.

Do you mean that -- instead of keeping both derailleur cables more or less parallel within the down tube of the frame -- the bike shop somehow crossed (or worse, tangled) them so that they were rubbing against each other!?

Thanks! I did a lot of thinking before working, that's for sure. Like chess. Yes, bad pro mechs are totally to blame for this. And I won't be using their services much longer, for sure. I jumped a big hurdle today, installing the cable and not being afraid to mess up my "baby". It's already not working right, get my buns in there and fix it, since 3 different shops couldn't do it. Not to mention the money I spent, I totally didn't need a new cassette or a freehub. Chain, yes, mine was 7 years old lol. Not salty about the cassette, honestly, I don't mind having the newest Ultegra version of it. I'm going to spend the money on the right tools for basic maintenance, and do it myself from now on. You got a list of the basics I'd need?

Yup, that is what I mean. I think it's possible, maybe not likely. I do know this. I took the old cable and put it alongside the bike, like measuring it out and finding out approximately where the most wear was happening. It was at the bottom bracket, where that donut was missing. From there to the RD, it was getting messed up. Glad I realized that piece was missing.

Does your bike have internal derailleur cable routing? Still not sure what you mean by donut; are you referring to a cable guide? If a cable guide is missing and the bare (i.e., not within housing) derailleur cable was rubbing against the frame or something else, that could definitely trip up the shifting.

Yes, it has internal routing. By donut, I mean "frame protector"? It's those tiny round rubber things that the cable goes through to keep it from rubbing on the frame. The cable goes through the downtube, through the cable guide underneath the BB, and then there is a hole in the chainstay where the cable routes to the RD. That hole needs a protector, it was gone, and the cable had been rubbing on the frame at that hole. I noticed there was also a definitive kink/bend in the cable where it entered the downtube, underneath the stem, not sure if that is normal or not but it doesn't seem like it would be.

I assume you have a repair stand. If not, get one.

I use a Feedback Sports Ultralight because I don’t need to support a lot of weight.

Lightweight. Stows in a bag. Takes a minute to setup anywhere. Ease of use is important to me.

Not the cheapest, but has worked well for 10 years.

John