For the wheels, grab something that has cartridge bearing hubs. Many cheap loose ball flip flop hubs use undersized bearings so they wear out quickly. If you want cheap and solid, Weinmann DP18 rims are quite commonly built into wheels with flip-flop hubs, and tend to hold up well to the abuse the average SS/FG rider dishes out. They are not light however. Any local bike shop should be able to get you a set, or help you figure something out a bit more lightweight if that is important to you.
Chances are your bike has a nice enough crankset that it has replaceable chainrings. If it uses 110mm, 130mm, or 144mm BCD, you could get a new chainring, a set of single-speed chainring bolts, and swap the bottom bracket out for something that gives you proper chainline. Finding some other used crank arms with a common BCD and doing all that is an option too. If you get an entirely new crankset, Origin8 cranks are cheap and not completely terrible. For something a bit nicer, Sram S-300 or Sugino RD-2. 1/8" chainrings must be used with 1/8" chain, 3/32" chainrings can be used with 1/8" or 3/32" chain.
Cheap single-speed freewheels are usually garbage, the internals do not handle the forces a strong rider can put out. ACS Paws is probably the best out of the cheap options. Shimano used to make solid single-speed freewheels, but it seems like the pandemic brought their quality downhill, so I would only grab Shimano if it is NOS. Not a lot of experience with them, but Origin8 Hornet 108 appears to be a solid choice. If you want to be absolutely certain your freewheel won't be a problem, White Industries is the way to go. 1/8" freewheels need 1/8" chain, 3/32" freewheels work with 1/8" or 3/32". Going all 1/8" with the chainring, chain, and freewheel/cog will help a bit with longevity, 1/8" chain with 3/32" chainring and freewheel gives a bit more wiggle room to work with a chainline that is less than perfect.
Not sure that the brand of fixed cog matters that much, but don't cheap out on the lockring. Shimano Dura-Ace, Surly, All-City are decent choices.
If you have room to move your brake pads down 4mm, then converting to 700c should be no issue. If there isn't room to move them down, you will need longer reach brakes. From what I remember, a 11.8 should at least be able to clear 32mm tires, and might even have room to go a bit bigger.