Ftw go electric, I've had the flexalite Black magic extreme fan on my daily driven 93lx for 19 years so far.
Oh by the way I live on the surface of the sun( nevada)
New or old pump shouldn't matter, full or empty either way the ecu sends power the pump to prime then shuts it till the engine is running.
Your constant running is somewhere in the wires ,ecu, relay?
I can't choose what rear you need without more info, but I can tell you I'd buy a used 8.8 fox body rear end and rebuild it with new traclok ( fords posi)
And axles with your gears of choice. Get the five lug axles with correct length and concentric for cobra 11.625 rotors ( got mine from LMR)...
Growing up in New York we named our cars and also threw change into the cars of friends and family when they first got there car.
Since my 93 is white I named her Milky. But after I work on her she's more of a "dirty white boy"