General comments here. 1) Horizontal dropouts like yours stuck around for so long in part because it made the frames easier to build and align. Only the vertical is critical. Variances in the horizontal can be easily taken up with the adjustment screws. So, yes, using those screws to correct for minor alignment issues with the tire at the chainstays is to only OK but even (at least in the old days of hand built for profit) expected. In fact, no one expected that a wheel would sit straight with the wheel pushed all the way back, no screws or with the screws in all the way (as you have done).
2) Dish. Flipping the wheel is a good way to see it IF you can seat the wheel reliable to be the same inside the dropouts either way.
2) OCD (not OLD! Obsessive Compulsive Disorder vs Over Locknut Distance) Small variances in either dish, frame alignment or both can add up to visual issues that will drive you nuts but if the bike rides fine and rides no-handed straight on level roads and no wind; it simply doesn't matter except between the rider's ears.
If I were in your shoes, I'd first check dish very carefully. Note how far off the rim is from center between the locknuts. (Say 3mm.) Next, I would put the wheel in, pushed back against the screws. Sight along the tire centerline. Does it line up with 3mm off the seat tube? Adjust the dropout screws until it does. Now, sight again, but looking beyond the seat tube to the head tube. Same 3mm? Good, the frame is overall straight by crude check. If not, probably time to run a string check which I will not describe here but is easily found. And note where the rim sits between the seatstays. This is governed mostly by the height of the dropouts, something you cannot change without moving metal and your wheel's dish. As long as the wheel sits roughly near the center, I wouldn't sweat it. (You never see it riding; just those riding behind you.) Adjusting brakes for out-of-center rims is usually very easy.
Dish - yes, perfect, centered wheels are nice but a few mms are very hard to detect riding. If your wheel has dish, think long and hard about the need to remove it. (I'm using "dish" in the context of the rim being centered on the bike, not where it sits relative to the hub flanges.) Re-dishing a wheel looks simple but if some spoke nipples refuse to turn, it can get messy fast. You are dealing with fully tensioned spokes. Re-truing will be needed, How much depends on your accuracy with your turns of the nipples and how well the wheel was built to begin with. You are doing all of this at full spoke tension where rounding off spoke nipples can happen. (Start with some new ones of the correct threads on hand.)
So, in short, perfect dish is the best but less than (if your brain can accept it) is often the much wiser course. Now, when you build/get new your next wheel, get that one right. (And in the meantime, don't bend or file your frame to correct issues caused by your current lack of proper dish.)