Re:
If you search around, you will probably find all sorts of paint strippers described as the best, but I found that Wilkinson's paint stripper was absolutely fine - just follow the instructions. When the stripper had been given enough time to do its stuff, I took the paint off with a shave hook. Usually, there was some undercoat left after that, but it came off with wet and dry fairly easily. I could probably have left that on, though. The angles where tubes meet are the only fiddly bits.
Respraying sufficiently well to protect the frame and survive knocks is the tricky part and you could end up spending as much on spray paint as a cheap powder coater might charge. You will need undercoat - possibly two different types, depending on the final colour you want - spray of the colour you want, then lacquer. And probably more than one can of each. Because you have to spray from a distance, there's a lot of wastage.
Durability is the issue, then. Rustoleum might be worth a try (as mentioned above). Spray paint designed for alloy wheels should be pretty tough but I've never tried it to confirm that.
Good luck with it.