95 Buick Century Rough Idle - Battery??

This is my first time ever posting about my car!

My grandpa's old wood paneled station wagon is sick. The idle seems to be getting rougher and rougher with every time I kill the battery (I've somehow left my headlights on four times in the past two months and needed a jump). It's to a point now where if I don't throw it in neutral at stop lights and keep a tiny bit of gas on the engine it stalls out.

Notes:
*When it first gets jumped after being dead, if I stop even for a second, it dies instantly.
*While cruising - it is fine.
*It idles smoother stopped in neutral than it does stopped and in drive.
*It seems a bit rougher with the engine is cold, but still rough and can stall with the engine hot.
*Some random times it idles just fine (like for example the first few seconds on start up, and then it sounds like the engine is decreasing in rpm automatically, and then it is rough again).

Could the dying battery and rough idle be related?? The battery guy says "if it starts fine, the battery is fine" (it does start fine, although keeping the radio on for 20 minutes without it running will kill it).

Also other problems with my car in case this has anything to do with it:
*I have a leaky tail pipe
*my ABS has recently been randomly engaging at low speeds (next problem to work on...)

Ideas??? Thanks!

There are several things at work here.

1st, when you kill the battery the ECU is losing its memory. This will make things like idle and such a little wonky until it relearns itself.

2nd, killing the battery several times could have in fact weakened it. It being in a weakened state could cause all sorts of electrical issues.

3rd, the fact that running the radio for only 20 minutes kills the battery tells me it is weak. You need to have the battery tested at some place like Advance, Auto Zone, etc (whatever parts store is in your area) They will be able to tell you if the CCA's are within range and if there is proper voltage.

Awesome!! Thanks so much. Still working on it.

Update: I just took it on an hour drive, and the last few minutes when pulling into my driveway the power steering went out, the battery light came on, but the car kept running (poorly, and the breaks didn't give out like it does when the whole thing dies)... only the steering went out!!

I'm not driving this thing again until I fix it...

You might want to have your alternator checked while you are at it. A weak alternator can be a burden on your battery and the rest of your electrical system.

That's it! The alternator froze up and broke my belt. Thanks so much! Any idea why an alternator freezes up? Just old or something else wrong? Also - do I need a new alternator, rebuild, or maybe there is something I can spray on it orrrr something (the car is a POS, cheaper fix the better...)? It was stuck and I put some force on it and now it (not smoothly) turns.

It all depends on the cost. If you have a local place that rebuilds them and it is significantly cheaper than a reman from a parts store, go for it. It's what I did with my Stealth when the alt crapped out on it. I saved myself over $100 at the time.