not sure if those chainring bolts are for single speed use hard to tell by the picture, and I don't know about the wheel, but how far wrong can you go for 50 bucks?
How many gears does the bike have at the moment? depending the spacing might be off, but thats solvable via a couple different methods.

How many teeth you want is a question only you can answer, I would suggest going out for a ride and trying to the gear that works best for you for all around use, it depends on your riding style, strength and the local terrain.

Make sure when you get a cog for it you note wether its 1/8" or 3/32" this will dictate what chain you want, considering your using a 3/32" chainring (if you reuse your crank) I would suggest a 3/32" cog, they can be mismatched as long as you go with 1/8 chain but it won't give you any noticable performance increase and will be *slightly* heavier (going for 1/8 cog and chain that is)

Edit: lets go back here a bit, does your bike have horizontal dropouts?