Logging of fixed base simulator time (737)
 

How do I know what abbreviation to use to identify the simulator used for this, is there a document somewhere?

Also, what is the code for the full flight sim, just FFS 737-700 or what?

There should be a number on the actual sim and tech log and you write that in your logbook.

You log it under the column "Simulator", I personally do it as you mentioned, such as 737-700NG whatever, I also make a note of the location. The sim does have an identity but you shouldn't need it in your logbook.