'88 Ford Escort LX not starting

Hi 'all, newbie here. I found this great forum out of desperation.

The day of Christmas eve, my car wouldn't start. Just a click, then nothing. It took nearly half an hour to jump start it. I let it run for a good half hour before taking off to run errants in town. I only got to make one stop, as... again... the car wouldn't start. Not a sound. Even a jumpstart didn't do it, and I ended up towing it to my house.

I recharged the battery overnight, and it did start the next morning. A strange sound prompted me to shut off the car within less than an hour of getting it started. The water in the radiator holding tank was boiling! Needless to say, the car hasn't started since.

A friend installed a new battery for me today and... nothing! Not a click, not a sound, nothing.

Something seems to drain the battery. How a drained battery and a boiling water-holding tank tie together I don't know and I'm wondering if I'm at the point of no return. I've just invested several hundred dollars for minor body work, a new tie rod, new snow tires, oil-lube-and filter etc. Before I tow off the car and get confronted with an ungodly estimate, I was wondering if anyone out there has experienced anything similar with an Escort.

If it helps any, there were a couple of odd occurances I've noticed:
1. The annoying door buzzer stayed on after the key was out of the ignition. To stop it, I would have to re-insert the key and turn it about a quarter turn or so before taking the key out again.
2. Two nights before the car broke down my stereo quit working while the power was on (until I find a new grounding place, the stereo does act up). Anyway, it wasn't until the next morning that I discovered the stereo face light was still on! I immediately undid the grounding and the light went off. Don't know if any of this means anything.

Any advice at all would be greatly appreciated!

Thanks,
notoolbox Chris

This is a follow-up. Turns out that the NEW battery I had installed yesterday was not charged! So, in a nutshell, looks like the car can be started with a freshly charged battery, however, it will not hold the charge very long. Plus, the water in the radiator overflow tank overheats.

Was not able to try anything new today as we got hit with a hefty snowstorm and I barely get to it.