Great. Your button problem is on the CPU board. There's no playfield input (you disconnected it) but the switches still register as closed.
F3 = Lower left flipper EOS (LLEOS)
F4 = Lower left flipper opto (LLFL)
F7 = Upper left flipper EOS (ULEOS)
They're actually not related. The EOS is playfield and the opto is cabinet.
01_flipper_direct_switches.jpg
The schematic. You should use a logic probe to determine where the signals are breaking down. If you don't want to or are unable to use a logic probe and you want to shotgun then the problematic ICs are indicated. Don't swap them all at once. Swap them one by one to reduce potential variables of change. Note also that Williams schematics are known to have errors so you should double check the schematic to board continuity. Odds are the schematic is correct but I have seen errors in them.
02_flipper_input_schematics.jpg