Foreword: I have very little knowledge about cars and how they work. Also sorry if this is long, I want to include every detail I can because this isn't making sense to me.
So a few months ago I'm driving along in my Cougar when I notice my stereo is starting to get quieter. A few seconds later my windshield wipers start slowing down and my window is fogging up (the defroster stopped working). I try rolling the side window down and up and find that is also very slow and now they almost stop working all together. I turn around immediately and start heading home. Halfway back, the car starts shaking a little bit, and is having trouble when I try to give it gas. It gets so bad I have to pull into the nearest parking lot where it promptly dies on me. I am then unable to restart it, so leave it there while I get a ride back home.
6 hours later I go back with my dad to get it towed somewhere, and I try to start it again. This time it starts and I am able to drive it home where it dies one last time. All the while it was shaking and stuttering for whatever reason (I actually thought it wasn't going to make it).
Basically it's been sitting in the driveway until last week when I finally got a multimeter and tried to figure out what was wrong on my own. At this point the battery was completely drained, even though it was very recently bought. I hooked it up to a portable battery charger and waited until the meter read 11.90V and put it back in the car. When the key was in, the dash lights did come on, though dim, although it wouldn't start. It wouldn't even turn over. I tested the battery again and this time it read 9.80V. I started to think it might be an alternator problem. So my dad and I tried to jump it with his car. The lights came on again this time they were the proper brightness but still no turning over. Then my dad took the negative clamp that was on my car's chassis and clamped it to the negative terminal of my battery. I always thought that was something you weren't supposed to do, but my dad said it was fine and I trusted him (note: He doesn't know much about cars either but more than me). Lo and behold, my car started. It was shaking a little bit, and as soon as I gave it gas it died, but it was something. I checked the battery again and it was still 9.80V or something close enough to that.
Now I don't really know what the problem could be coming from. If I try to logic it out, I think it could still be a problem with my alternator, and his car was just able to keep my battery charged until I tried to give it gas. Am I close? Or could it be a problem with the starter? Or even just the battery cables being lose or frayed? It's a pretty old car with a lot of miles but it's been very well taken care of, both my myself and the previous owner, and it's a 2.5L Manual if that matters any.