Alright. So today I dropped off the 105 derailleur I had. The shop worked with a new 10 speed ultegra chain (the only type they carried) and they managed to tune up the shifting perfectly. I also checked the hanger and the original shop seems to install the correct type of hanger (although I don't know about the length of it).
1., After the initial rear derailleur replacement, the shifting was bad because of the damaged chain (and probably recycled rear derailleur).
2., The second shop put on a SRAM chain as a replacement and failed to tune up the drivetrain properly.
3., The last shop tried to tune up the drivetrain with the SRAM chain and tiagra rear derailleur, but they couldn't do a perfect job. The reason they wasn't able to get it right with a new tiagra rear derailleur and new whatever chain they had is a misery.
The take away I ended up with after talking with the mech at the shop:
x) Sram chains usually work well with Shimano drivetrains, but might effect shifting and they recommend to use shimano chains with shimano derailleurs.
x) It pays off to return to a trusted shop, because at the end they managed to figure it out and tune it up just fine.
Thank you guys for all the suggestions and help. I really appreciate it.