Placement of varistor with respect to fuse
What risk am I running if I put the varistor before the fuse?


You risk burning out the varistor, the varistor is built to short circuit in the short term not the long term. In the event of an overvoltage condition, the varistor turns on and shunts the current. If the current is shunted for too long the MOV can heat up, the fuse protects the MOV.

Although MOV's can be said to be somewhat bulletproof, no component can handle the full 20A+ from mains (or a transient event)


  When subjected to stresses above its ratings, an MOV can fail as a
  short circuit. If applied conditions significantly exceed the energy
  rating of the device, and current is not limited, the MOV may be
  completely destroyed. For this reason, the use of current-limiting
  fuses is suggested.


Source: https://www.littelfuse.com/~/media/electronics_technical/application_notes/varistors/littelfuse_the_abcs_of_movs_application_note.pdf

Another thing that is noteworthy is MOV's are only rated for X number of transient events, in the 10's to the 1000's. After that the MOV can fail as a short circuit, its nicer to have a fuse to blow then lots of smoke. 


  What risk am I running if I put the varistor before the fuse?


The varistor might blow and turn into smoke, I'm willing to bet this will not fly with some regulatory standards for power supplies. 


  What if I put a second fuse in series with the varistor? Does it
  negate the varistor's protection of the downstream circuit?


Depends on what current the fuses are sized for, but this could result in a situation where only the MOV's fuse blows and then the circuit is left unprotected.

If the object is to save space adding another fuse isn't going to help with routing.