'97 Plymouth Breeze intermittently starts

My Plymouth Breeze is having issues starting up. It started a while back, my wife was driving the car and she got stranded because her car wouldn’t start. I gave her a jump. I had the battery and alternator tested and the battery was bad, but supposedly that alternator is good–so I replaced the battery. We get a week or twos use out of it, and the the rear brakes sprung a leak-- so I replaced the rear brake lines. When I tried take the new brakes lines for a test drive the car wouldn’t start. I have a battery charger and used the 50 amp start feature, the car would still not start (it would turn over but not start). So I gave the battery a deep charge from the battery charger and the next day it was fine. My wife used it for a week and all seemed well. That is until the other day where she tells me it has a new problem. Apparently the defroster randomly turns off and on, and the vent constantly switches which direction it is blowing. She took it out for a drive while it was like this and got stranded again. I went to give her a jump and the car wouldn’t start at all. In fact none of the electrical came on, it was as if the battery had been disconnected. So we eventually get the car towed, and as I put the key in the ignition to put the car in neutral I hear the beeping to indicate the car door is open. I tried to start it and all of the electrical shut off again. So the guys tows car back, having to push it into a parking spot. I realized the tow truck driver left the keys in the ignition and it just occurs to me he had rolled the window down. I get in the car and turn the key and it starts up without hesitation, like there were not problems at all. Needless to say I’m a little confused and frustrated with this car. I’d like to get another year or so out of it before I replace it.

So here is what I am thinking…
Battery: This should be fine I just replaced it.
Alternator: I had it tested coupled with the last scenario leads me to believe this probably isn’t the problem, but is suspect.
Starter: This seems to work when there is power, so I don’t think that is it.
Fuses: They either work or they don’t, there is no “getting better” so I doubt it is a fuse.
Computer: I think this might be the culprit. Before I had to give it another jump (when it wouldn’t take) things were turning off and on at random. It is also the only thing I can think of that might act in a fickle way.

Do people agree with me, or is there something I might be overlooking?

Edit: Another theory is a loose wire coming off of the battery.

Check the connections from the battery with a multimeter, set to measure voltage. If you don’t have one, visit a HarborFreight. They have one for just a couple of bucks that will do just fine for what we need to do.

Measure across the battery. It should be 12V.
Move the negative lead of the multimeter to the chassis or engine but leave the positive lead on the positive of the battery. You should read 12V. If you don’t, you have a bad ground.
Move the positive on a fuse you know is alive by pulling this fuses out and measure on each leg. Negative on the chassis/engine. On one side of that particular fuse’s connection You should read 12V. If you don’t, you have a bad lead from the battery to the fusebox.
/Carefully/ measure the fat cable on the starter and ground. You should have 12V. If you don’t, that cable is likely bad.
If it passes the above test, try it again with the key in the ignition and trying to start it.
There’s more but that’s where I’d start.

Btw, giving a car a jump causes all sorts of weird spikes and brown out voltages (ie higher and lower than normal voltages) to be applied to the ECM.
Much like a computer when the power browns out right before you lose power, it can do weird things. It may be a red herring.

I have experienced the blinking defroster light, the body computer resets when power is restored. The problem was a loose battery cable. Check the battery connections and the jump start junctions under the hood. Sometimes people leave the negative jump start cable junction loose.

@RemcoW Before I read your comment I had called my Dad and asked him, he suggested the loose cable. I tried to start the car again, and again no power. I took my multimeter and read 12V across the battery but not across the post that goes to the fuse box (On a Breeze the battery is behind the front driver side wheel and a cable runs up to the hood and immediately connects to the fuse box). I tightened both the positive and negative leads from the battery and retested and the post to the fuse box, it read 12V.

After reading your comment I went back out and measure the voltage from a fuse and at the starter, both read 12V.

@RemcoW & @Nevada_545 It is good to know that the loose battery can cause the weird behavior my wife observing with the defroster light, and likely the AC Vents.

After the responses and the testing it seems to be a loose cable, thank you all for you help!

Edit: Oh, forgot to mention. When I tightened the negative terminal it was pretty loose, so that was probably the culprit.

Okay, now try to start it with that very same multimeter in place.

Edit: Oh, so it runs now? Good. Keep that multimeter handy, in case you see it again.