This looks awfully close to your bike, the 1987 Iseran, USA model:
I'm pretty sure the yellow-orange-red "rainbow" markings put your bike in the 1986-1987 time frame. The Iseran does have Shimano shifters spec'd, but the derailleur isn't specified in that link.
Your seat post also looks different from the seat post shown in the brochure image, ...
You will probably struggle to get a perfect ID on this bike unless you get a response from someone who used to sell them. I have tried to identify Peugeots in the past and there's only so far you can go. The naming scheme is difficult to work with as the model number the bike was sold as is never printed on the bike.
The tubing isn't the very best available ...
I have a peugeot frame of a similar age and have found it to be more bother than its worth.
There was a period when peugeot made an effort to make all of their compenent dimensions round metric numbers. This practice was not adopted by many other manufacturers so getting parts to fit that frame might be a hassle.
On my peugeot frame;
Steering tube internal ...
Following on to Andy P - agreeing that it's a Monaco but the German version.
It looks like a 1988 P 8/X Monaco. Peugeot's touring bike offered in Germany.
The parts (Center pull Weinmann 750s and Shimano Positron derailleur. Fenders and rack removed.) and frame construction match.
The bike in the original post is a different color. Since I can't read the ...
This is really an extension of the answer from @AndrewHenle - I just extended his detective work.
I believe the bike is a 1988 'Monaco'
I found a paint scheme that matched the OP's bike on the 1988 'Bordeaux':
However, this features down tube shifters, so is not the bike in question.
Also from 1988 is the 'Monaco'. The picture below shows it in white/blue, ...
A large vice can be used to clamp the cup and the frame should provide enough leverage, especially since you've almost completely disassembled the bike. Use some penetrating oil from the inside of the BB. Give it some time to seep into the threads. A hair-dryer could be helpful but a hot air gun might damage the paint.
French BBs are usually right/right and ...
Answering your numbered questions first:
Will the rusty/blank spots be a functional problem down the line?
A: Bare steel will rust. It needs to be covered in some way to prevent corrosion.
Will power coating work well with the steel frame?
A: Powder coating is an option that will work well on a steel frame. Here is a Sheldon Brown article that talks about ...