The way to know for sure is to borrow a 700c wheel set and test fit them to your frame. You have to test fit both front and rear because they might not be the same. Modern double pivot caliper brakes (look for a long arm version) will definitely be better than what you probably have now.

Unless your bike was designed with a bunch of fender clearance in addition to having 27" wheels, I'm betting it will work.

The paint thing is a personal choice. A modern powder coat job for $100.00 - $150.00 or so will give it a classy resto-mod look. Keeping the original paint keeps the bikes classic patina. Other people can argue endlessly which is better but it's your bike so you hold the only vote that matters.