OK, to take you your question in order:
1. Tyres. 27" wheel tyres are readily available, anything from 1" through the classic 1-1/4" sizes. If nowhere else, the local Performance Bike shop always seems to have a half decent selection in stock. 700c wheels are no particular advantage for general use, unless you're in an area where 27" tyres aren't readily available.
2. Wheel condition. I'm going to assume that your appraisal of spoke fatigue is entirely visual, in that they don't look new, but there's no broken spokes. If this were my bike, I'd put on a new set of tyres and tubes, and then go take it on a 5-10 mile ride. Keep it gentle, no extreme downhill screamers, etc. Then check your wheels mechanical condition. Odds are, they'll still be fine. If so, just clean the up - chrome polish will clean spokes up nicely. If a spoke or two does snap, replace them, then try the gentle test ride again - if you have more spokes snapping you've got a problem, although I'll bet this doesn't happen.
3. Chain and cassette. I always replace the chain when I get another old bike. 6-speed (that's what they're called nowdays) chains usually don't cost more than $10.00. As to cassette, I have a feeling what you've got is a freewheel, not a cassette. They're easily available on eBay. Go for a Shimano as they're easily available, the quality is always good, and the removal tool is the easiest to get. Park makes a good one. As you've got a Japanese bike, the freewheel is industry-standard English thread.
Just the same, for that first gentle test ride, spray some lubricant on the old chain and into the freewheel body. As long as the latter clicks and freewheels properly, you're probably not going to have to replace it unless the teeth are really worn - and I'm talking physically worn down, not just a coating of surface rust.
When I got my Raleigh Gran Sport last year at the motorcycle show, I rode it that day to get around the show - it was my reason for buying it in the first place. $4.00 sure beat walking. There was corrosion on the cable guides, so it wouldn't shift well, and cosmetically it was what is called in the motorcycle world a rolling basket case, but I put it in third gear and it got me around the site for the entire day, by which time I'd figured out just what I'd needed to do to get it into good condition. And that included the original tyres (cracked gumwalls) and tubes (they held air).
Never underestimate what an old beater bicycle is capable of. Just the same, make sure that first ride is gentle and at low speed, so you don't get yourself in over your head.
Syke
Deranged Few M/C