So very, very, very confused...again.

I'm not sure you'll find "non-friction" shifters for a 5 speed setup. Your cassette is 5 speed, so the shifter theoretically should match (and be 5 speed), but if that Suntour unit is really just friction rather than index (click stops for each gear) you may be ok. Don't know whether it's called "6 speed" just to indicate it has enough range of movement to cover that.

You will need a rear derailleur that matches the shifter (broadly speaking). The derailleur movement is matched to the cable pull the shifter does, and they need to be compatible. Assuming it's friction shift, if the shifter is not a "match", you'll get either not enough shifter range to cover all the gears (because the derailleur doesn't move far enough for each cable pull per shift) OR the lever position will be way too sensitive to be useful if the other way round..

Short cage *might* be fine, but remember that back when that gearing was the norm that a lot of race "cassettes" were crazy like 12-21 or similar, so a 15-28 is relatively wide range for the period. Finding a rear derailleur that's in working condition for that era to suit may not be easy... A top gear of 15 is pretty short. Would have been for a touring bike rather than race I expect, especially with a 28 bottom, so look for a rear derailleur more aimed at a touring bike than racer.

Lastly, does your wheel set and frame accommodate this? The spacing between the rear dropouts required for a wheel with a 5 speed block was typically 120 or 126 mm I think, while anything from 8 speed onwards was 130 (and now most are 135 or 142 with disc brake stuff). Your wheel free hub will need specific tools too I'd guess.

If you just use the 42, no need to remove the 53, because if you do, you will need shorter chainring bolts to keep the 42 in place. Which is not a big deal, but has to be done.

With a 42 front, I'd use the 16 or 14 at the back, unless you live in a really hilly area. Probably the 16 would be my pref for all round use. My last single speed roadie was 46 * 15 and it was great on dead flat terrain, but any hills at all quickly got to be hard work.

Does your frame have the horizontal dropouts with the tension adjuster screws? Easy to do single speed if you do. Modern vertical dropout frames are much harder to convert to SS and get lucky with chain tension.

The chain size is still pretty much the same pitch (link length) it's always been. The difference is that the more gears you have, the thinner the chains are (side plates and pin roller bits) to fit in the narrower gap between cogs. You can buy a 5 or 6 speed "size" chain pretty cheap.