The Gods that govern serendipitous bicycle discoveries have smiled on you-(I bet it's your size,too) and left you with a time/money/skills/dedication equation to negotiate, as they are wont to do.
IMO on a thirty-year-old neglected bike, every component should be dismantled, cleaned, inspected, lubricated, reassembled. It looks like you've done most of it already, but I notice you haven't got the cranks off yet, so if you haven't already, and are able to, meet Lady Luck half-way by investing in a crank-extractor and a set of headset/BB spanners something like these:
Regarding paint and chrome, it depends on whether you want it to look like 'never left the showroom', or 'once neglected, now cared for'. I favour the latter, and would aim to preserve as much of the original finish as possible... not only for aesthetic reasons either. Others might favour blasting, repainting, new transfers, rechroming, etc.
Personally, I would strip it right down to the frame, (including removing BB, forks, and headset), and work on the rusty bits with a wire brush, fine wet-or-dry paper, etc, removing all flaking paint and chrome, and, once you have assured yourself that the tubes haven't rusted through anywhere, touch up with primer and paint. I would also clean up inside the tubes as far as possible.
I don't think anyone actually replaces rusty rim-eyelets (I may be wrong), but they can be cleaned up. To do a thorough job would entail removing the tyre from the rim and the rim from the wheel. If they are rusty on this side, chances are they are rusty under the tyre where the spoke-nipple seats as well. If it's the rear wheel get the sprocket-block off first. Clean up the rusty eyelets with wire-brush, etc. those stainless steel kitchen scourers are handy too. Satisfy yourself the eyelets are still structurally sound. The 'inside' of the eyelets, where the spoke-nipple seats, could be treated with some anti-corrosion product and/or painted to help prevent further oxidisation, obviously you don't want paint on the surface where the tub is glued. Then rebuild the wheel and true it. I guess if you don't want to dismantle the wheel you could do one eyelet at a time, removing the spoke-nipple, get the end of the spoke out of the eyelet, clean up the eyelet, replace the spoke and nipple. You would probably still need to true the wheel once you've done all the eyelets. If you do one at a time, it might be advisable to slacken off all the spokes a few turns first anyway..
Others probably have more efficient methods..