What do I need to replace my singlespeed freewheel cog?
Park tool has a guide to figuring out what type of free hub/freewheel you have, and what tool you need to get it off. 

https://www.parktool.com/blog/repair-help/determining-cassette-freewheel-type. 

With that knowledge you should be able to to figure out what type of replacement freewheel/spocket you need (or two, as you have a flip-flop hub). 

Update:

Also see the Sheldon Brown page on Freewheels.

Depending on what the freewheel thread turns out to be, you should be able to get a replacement freewheel that has a more standard lockring. 

You might consider creating a 'bodged' tool to get the current freewheel off. You might be able to 'drift' the lockring loose by tapping the ring around with an old screwdriver and hammer.