*** Final Edit *** Solved
One of the wires going into t4 on the Fan Control Module was rubbing against the frame and shorted the internals of my after-run coolant pump.
I fixed the wire that was rubbing against the frame and replaced my after-run coolant pump, and my TT has been running like a champ.
*** Edited ***
As stated in the title, Fuse 16 keeps blowing in my 2001 Audi TT Quattro Coupe 225. The fuse basically powers the Fan Control module, so it's used for the magnetic clutch on the A/C compressor and the after-run coolant pump on our cars. It also supplies power to the G65 A/C pressure switch (this can be confusing, as the VCDS scan will always say F129 switch open circuit - G65 is a newer version of F129 with three wires instead of four).
I've checked continuity from the magnetic clutch to the Fan Control Module, and it's fine. Same for the after-run coolant pump and the G65 A/C pressure switch.