The drivetrain wears together, so when I replace one piece I go ahead and replace everything. Since you're doing the chain and freewheel, you may consider replacing the chainrings as well. If the chain is less worn than the freewheel, it may skip: vice-versa.

You can use a wide freewheel, but it's not necessary. An advantage of a wide freewheel is that it would be quieter. Same for the chainring. Before buying a chainring, count the number of bolts and measure the BCD. Wide parts are usually advertised as Wide, but could also be advertised as Single speed or BMX.

Once you have all of your parts, cut the chain to the correct length and install them. Use the tensioner to eat up the length of the link left over after cutting the chain.

It may or may not be obvious that this will require specialized tools. Good luck!

since no front derailleur you just need to find a single speed freewheel that can accept your 1/2" x 1/8" chain. if your current chain ring on the front is also compatible with the chain you don't need to replace it. you will need to ensure the chain line is straight so the wider freewheel may be neccessary