2004 Mazda3 battery died/leak/shaking

I drive a 2004 Mazda3 4 speed automatic, 2.3L that's at around 80,000+ miles. I had my battery replaced approx 2 years ago and my alternator replaced last year.

So yesterday, I leave from work, car is fine. I run an errand and on my way back from my errand, I notice check engine light is on. Don't know if this is helpful, but I'm also driving on fast highways on my way back from my errand. Then I'm on local roads, stopped at a light. Light goes green and I start to accelerate, but suddenly it's like my car seizes up, all the lights on the dashboard go on, the steering wheel goes stiff, the break pedal wont budge, car breaks down. I guess, the battery died. I figure crap, the alternator, AGAIN? When the tow truck comes - didn't trust my car to bring me home with a jump - he notices that the cover on the bottom is loose and I have a major leak. He asks if I hit something, but I really don't recall hitting anything. Though I did remember that I heard a suspicious low rattling sound very briefly while on the highway, but I heard it very briefly.

Later my friend jumps the car to check it out, engine is running fine while in park, but as he puts it in reverse or drive the car starts to shake and powers down.

Some additional info about my car lately:

Check engine light was on for awhile a few weeks ago. When I went to AutoZone, I was given a bunch of different codes mostly regarding some sort of air leak, something about an oxygen filter, or a possibly faulty valve of some sort (none of this means anything to me). I wish I could provide codes, but I've lost them since, sorry. Someone suggested I try replacing my gas cap first. I did so and the check engine light went off. Most of the time. It would come on again from time to time very briefly, but mostly it would stay off.

I should also add that lately when I'm at traffic lights - especially after exiting off highways after really long drives - my car has been shaking. I thought the check engine light was due to that, but since it went off, and the shaking wasn't happening all the time, I sort of let it go.

I know close to nothing about cars, so if there is anything I need to check to provide more info, if you could give thorough info on how to go about doing that, I would much appreciate it. Thank you