Plumbing – Gas hot water tank quit working and leaked

I would replace the P-T valve regardless, as it sounds questionable. They're about $10-15, plus usually a pipe directing it to the floor or even a nearby floor drain (which you can probably re-use).

It's a safety device that prevents your tank from over-pressurizing (which in very extreme cases can actually cause an incredibly powerful explosion -- check youtube for some entertaining mythbusters clips). You also don't want it to trip accidentally, as it will pour a lot of very hot water out and will not stop until the P-T valve shuts itself off automatically (unlikely if it's broken), or someone shuts off an upstream valve. If your floor drain is clogged, or gets overwhelmed, you have a flooded basement/house.

Your expansion tank has likely failed or isn't large enough for your hot water tank. Expansion tanks have a bladder in the middle that ensures it is half full of air and needs to be periodically checked to be sure it's properly pressurized. These bladders eventually fail, allowing water to enter the air side, which eliminates the ability to absorb the expanding water.

Replacing the expansion tank may be a simple process if it's just screwed on. With the water shutoff, you unscrew the existing tank, pressurize a new tank, apply some plumbing dope to the threads, and screw the new tank on.