Can I take this adapter, cut off the unused arms to save some room behind the dash, and seal it off with liquid tape?

Yup should work.
Just make sure none of the wire hairs touch on the cut pieces before you tape it up.
Heck you could cut off the ends you need, solder them up and loose the puck in the middle altogether.

That thing is sweet. Where'd you find it? I've never seen those before.

My brother picked a couple of them up when he was putting in a cd player for his work car, a lil hyundai accent. He asked if I needed any stereo parts and started pulling these out of a bag and asked if I wanted anything. I just called him and it was in a car stereo installation kit from Walmart or Target.

Cool. Thanks! I'm gonna watch for those.

From an electronic standpoint, that monster is a disaster. It defeats the purpose of "shielded cable" and allows for the introduction of all kinds of noise. Personally, I wouldn't let that within a country mile of my car. Except to maybe run over it a few times.

Well I figured I'd give it a try and if it doesn't work I'll give in and go buy one... First I cut all the unnecessary plugs off then cut off all the rubber surrounding the puck. I trimmed off whatever I didn't need of the guts then sealed it all up with the liquid tape.

By industry (or any reasonable) standard this is completely unacceptable when dealing with RF signals. Personally, I don't even feel this should be posted since it may only encourage others to follow what is an extremely poor practice.

Isn't it better than the 6 legged creature that it started it'* life as?

Not by much. Splicing coaxial cable requires specific techniques and parts to ensure the shield remains uninterrupted. The intial product was open not only to noise, but with all the "attachments" would likely have changed the impedance. Although you can't measure this with a multimeter (since it will appear as an "open"), coaxial cable acts as a load within an RF circuit. 50 or 75 ohms is quite common in many audio and video applications and you'll generally see these values printed on the cable itself. You have a splice in the cable and as such the shield is interrupted. When dealing with RF signal handling, this can have a profound effect on signal strength and quality. The degree of effect will vary depending on frequency range, length of the interrupted segment, proximity to noise sources (which in an automobile is extensive) amongst others. This may work for you, but by no means is considered acceptable practice. To do this "more correctly", you would need to cut that disk out of the centre entirely, crimp male connectors on each end of the bare cable and use an coupler. This isn't ideal either though because every connector in an RF circuit has a resulting signal loss. The better practice is to use the correct adapter with as few connections as possible. There should be no interruptions is the shielding. Absolute best scenario would be to crimp the correct connector on to the existing coax and not use an adapter at all. This isn't reasonable for most and therefore some compromise may be necessary.