When I turn the unit on, all I hear is static until I turn the squelch all the way down to silence it.
the way up? (More squelch = up, not down.) Usually one can achieve squelch at much less than all the way up.
I have never heard any conversation through the static. So, obviously something isn't working.
Possibly. I assume you've listened to channel 16 for a lengthly time?
How does one troubleshoot the radio?
Troubleshooting the radio, itself, requires knowledge and test equipment.
One item that looks like an issue is the connector between the deck and the mast for the antenna cable. It's a coax cable screw-in terminal and it looks slightly corroded.
That is bad. That could definitely be a problem.
Is there a way to connect the radio to a "makeshift" antenna in the cabin to at least test the radio?
Yes. They make emergency antennas. You should have one anyway.
Any specific tests that can be done with a basic multimeter to test the antenna cable? (Ohms of resistance to look for... etc...)
Yes. Resistance (DC) between the coax center conductor and shield, when the antenna's connected, should be very close to zero (a few ohms, at most, depending on coax length). With no antenna connected, it should be infinity.