Some multimedia sets you probably wouldn't even need an inverter. For example, my oldish Boston Acoustics set takes +12v from the wall adapter. If I were to use an inverter with mine in the car, I'd be going [car +12vdc] --> [inverter +115vac] --> [speaker adapter +12vdc] --> sub, which just seems really inefficient. If this is the case, you should (in theory) be able to just get some wire + matching power plug for your sub from Rat Shack and wire it straight into your battery. Problem with this is, you'd have to remember to unplug it each time you parked.
There is a switched lead going into the back of the stereo (the red one) which would allow the sub to turn on and off with the ignition switch...but I wouldn't run the whole thing straight off of it, as it is not designed to run all of that extra current through it (and you would almost certainly blow fuses). If you knew a bit about rigging up relays and such, you'd probably be able to use the power antenna / remote lead (the blue one on most decks) in conjunction with a relay to turn the setup on and off with the switch.
MAKE SURE you pay proper attention to plug polarity (going into the sub) and also be sure to put an inline fuse in the "hot" wire. Never have unfused connections to the battery for any reason.
Easiest way to get sound out would be to have an aux out on your deck. You can get cables that have RCAs on one end (connection to most aftermarket and some factory decks) that terminate in a single 3.5mm phono plug, which would connect to the sub. No crossover needed, since all of that circuitry is built into the multimedia speakers already.
Failing that, I guess you could get a line-output converter and tap into the rear speaker leads either behind the stereo or in the trunk, depending on accessibility. These tend to introduce interference though, and would probably be more hassle than it is worth and end up sounding like crap.
DISCLAIMER: I've never tried any of this and never plan to. I'm just sort of rambling to myself and writing stuff down. So I'm not responsible if you try any of this and cause carnage
And this is completely fucking ghetto
