Rough idle could be several different things. The simplest solution is replace the spark plugs and wires. The more complicated would be a bad O2 sensor, or the timing belt is old, and has jumped timing. If it’s the timing belt, you need to get it replaced ASAP! Most engines are interference, and if the timing belt breaks, the engine will be destroyed. I’d hope that a 2001 would have had the timing belt replaced at least one in its lifetime, as it’s a routine maintenance item, that is supposed to be replaced every 100,000 miles. Hope this helps!

Not familiar with how keyboards are attached to Apple laptops, but, I'm sure they can be taken out. This is the last thing before just buying another keyboard. Take it out, soak it in warm water with Dawn soap for an hour. Rince with hot water while scrubbing. Pat dry and then let dry in rice for 48 hours. After that, re-instal it. If it still doesn't work, it's not dirty, just worn out. This might sound stupid, but in the last 20 years of fixing computers, it's brought back around 60% of keyboards that were considered "A Lost Cause"... As for the battery, you need to get a genuine Apple battery.

Either it's full of data, or hasn't been properly formatted.

How old is the battery? Over time, rechargeable batteries lose their ability to hold a charge. It might just be time to replace the battery.

Charger what broke off? The plug? If there was enough force placed to break it off, I'd say the internal receptacle is most likely damaged as well. One trick I use to get plugs out is to get some Silly Putty. Squish it into the plug, then quickly yank. It might not come completley out, but it should get it out far enough to get some tweesers or needlenose pliers in there. Other than that, without a pic showing exactly what's going on I can't really give a more detailed answer.