Just wondering if I can cut the cables going to the two little speakers, and rewire two car audio tweeters to the input wires for the woofer. Will this burn my house down, or is it do able?

If your impedance is right you shouldn't have a problem, you do know what impedance is right?
;)

How does the woofer hook up? Does the woofer have an external amplifier? Or is it just a big box that has a 1/8" jack that plugs into your computer?

How does the woofer hook up? Does the woofer have an external amplifier? Or is it just a big box that has a 1/8" jack that plugs into your computer?
The latter. And I went for it anyway, and it sounds great. The speakers are rated at 25W and output is 12W, but it doesn't seem to be a problem, no way I'm going to have it cranked up in my house, I just want some quality tunes for when I'm trying to get to sleep.

If your impedance is right you shouldn't have a problem, you do know what impedance is right?
;)
Ohms, IIRC? And I couldn't see it marked on the tweeters (the woofer box said 4)

4 ohms is just about the most common for car audio. I cant recall ever seeing a car audio tweeter or midrange that was anything other than 4 ohm.

you won't burn down the house.
You may burn out the tweeters, in a similar fashion to burning out a light bulb. (poof/pop= dead).
You need to not send bass notes to the tweeter.
a crossover capacitor inline is the easiest fix.
without the capacitor or cross-over unit in place, don't do it.