How to run 12V compressor directly off battery?
"is this the correct size fuse?"


The mechanically correct size fuse is the one that mechanically fits the fuse holder.
   The electrically correct size fuse is the one that protects against burnout (According to the Manufacturer - 15 Amps).


  "...glass or ceramic and slow-blow or fast-blow. Which should I be using?"


Glass or Ceramic, it doesn't matter - even plastic would be okay.
   Fast-blow is for protecting sensitive things like diodes and transistors, not motors or light bulbs, etc.


  "Is it ok for the in-line fuse holder to be rated at 20A?"


The fuse holder rating indicates the largest fuse you should use, smaller is okay.


  "Is my diagram correct?"


While electrically your diagram will work, for safety reasons the fuse should be in series with the HOT side of the circuit as near to the power source "BATTERY(+)--[Alligator Clip]" as possible!!!