#1: Is the rear hub/gears interface a splined freehub/cassette or thread-on freewheel? See here for the difference.

If it's a cassette, you can get a single cog, several spacers, and a tensioner. If it's a freewheel, you can get a single-speed freewheel, then respace the axle, redish the rim (this is a more involved process and you may need the bike shop for it), along with a tensioner. (In the off-chance that your frame has horizontal dropouts, you don't need a tensioner. Also, the rear derailleur could work as a tensioner with the limit screws adjusted.)

For more info on single-speed, see here.

Considering what's involved in going to single-speed, you may find it easier and cheaper to just get a new chain, freewheel/cassette, and rear derailleur (RD).

IMO, the chain and cassette/freewheel are wearing/replaceable parts anyway, so the RD is the only part really being replaced, so I think it's worth it.

#2: I'd ask a bike shop how much to replace the rim and replace it to the existing hub, including new spokes likely (or change the entire wheel). You might want to get the fork checked out. IMO, this bike is worth replacing parts.

#3: Are there provisions for hand brakes on the fork and frame? Maybe a hole for caliper brakes? If you can install hand brakes, I'd be tempted to leave the coaster hub alone. To replace it, you'd need to relace the rim to a new hub (with new spokes likely), which is an involved process and you'll probably need the shop to do it.

It may be more economical to sell this bike and get something that better suits your needs.