Done this rescue a few times and most of them came back to life, so here's how I check one before it ever sees a car battery. First thing, pull the top cover and look inside. You're checking for a burnt smell, scorch marks on the board, or any can-shaped parts with bulged tops or crusty leaked stuff around the base. Dust is fine, blow it out with some compressed air. Corrosion on the board from a leaking roof in the truck is what usually kills these, not age. If the inside looks clean, you're probly okay to power it up.
On your bench supply, set it to 13.8 volts if it goes that high, and most old mobiles want 10 to 15 amps available, so check what yours can do before you connect anything. If it's a little supply that only does a couple amps, it may fold over when the radio keys up, but that won't hurt the radio. Start with the supply current limit turned all the way down if it has a knob, then bring the voltage up slow and watch the ammeter. A healthy radio sitting in receive should pull well under an amp. If you see the needle slam to the max the second you connect, kill it right away, that means a shorted part inside and you just saved your fuse block.
No mic needed for a basic bench check. Receive is the easy half. If you've got another local ham or even a handheld, key up low power from across the room and see if you hear it. Some radios will also show signal strength on the display when it hears something, which tells you the receiver front end is alive without transmitting at all. I wouldn't transmit on it until you know the previous owner didn't beat on it, but receive alone tells you most of what you want to know.
Mics are the annoying part. They almost never swap between brands, the wiring is different on nearly every one and some carry the mic audio on different pins, so a random mic can do nothing or can even short the transmit line. Find the model number on the label, usually on the back or bottom, and search that plus the word mic pinout. Somebody has documented it. Then you can either hunt a used mic for that model or solder up an adapter yourself once you know which pin does what.
Cut power lead is no big deal, just solder on new leads and mind which one is positive, they usually mark it on the board or the back of the set with red and black.