I plugged my Nokia into a new car charger, and it didn't work. When I disconnected the device, and pulled the charger out of the cigarette lighter, the charger was smoking! The charger was rated at 2.1a.

Now the phone turns off randomly, even when the battery indicator shows a charge. I bought a new battery, but my N900 still turns off after the screen turns off.

Anyone know of a fix, or is my beloved N900 fried?

This seems to be somewhat similar to your problem.
If your N900 turns off randomly, then it's hard to pinpoint exact place where hardware needs to be repaired.
If it turns off when the screen turns off, then it should be easier to solve. Maybe, there is a short-circuit of some kind related to display and battery charger?
I know of a computer with opposite problem: once the computer is turned off, its display (graphic card) becomes disconnected from the computer, and the computer needs to be warmed up to get it to turn on with working display. The hypothesis is that when it warms up, the metal parts become connected again.
Best wishes.

seems like your charger damaged your n900. this also happened to my cousin once he left his phone on charge for about a whole day and a half day. its two circuts were damaged by over charging.

Its phone was also behaving like yours. working fine untill screen turned off.

Was your cousin able to repair the phone?

I was using it as a 64gb flash drive with a GUI. It made up for all things my Nexus 4 lacks. (Storage, removable battery, storage...)

Somehow, it still could be a loose battery connection?...

What you have to do is, get a needle, and bend the 'pins' that are inside your batteries together, so when you place your battery in the phone again, the contacts will sit in tight

Hope that this will solve your problem!

I read about that on another thread. I shook my phone like crazy when it was on, but it never turned off.

Possibly on a related note, the processor indicator shows that my phone's processor is basically maxed out all the time. Even when there are no applications running, and is in airplane mode, it constantly flashes red.

Reflash it and check out

And how in the name of Zarquon do you suggest doing that with a fried USB port?

my knowledge is very crude indeed it can be simply done by fbus cable using the many suitable boxes
or soldering thin wires for a serial port
and many a times seemingly hw problems are just cured by reflash
alternatevly he can use a multimeter to check the fault and replace the component or rather remove it and make a jumper insted the charging components that usually get fried can be found from diffrent boards easily and most of them can be removed and a jumper can be placed reheating the charging ic and retru are also an option if one is unable to reball them :] detaching and replacing the usb post also does the trick sometimes as theres no clear way to check if the port got fried
thease are just the basics i do when i fry something

Just to give justice - OP never mentioned damaged USB ports, he just wrote, that *car charger* was smoking.
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Are your usb ports working? It's quite crucial information.