I think the answer you're going people telling you to avoid doing this more than helping. But I'll suggest some things.
Max power draw you can get from a cigarette lighter would depend on a few things, including the gauge of the wire ran to it, and the size of the fuse. I know mine lists it at 10 amps, of 120W max. So I wouldn't recommend using your 150W inverter off of it.
You would want to ground the inverter as close to itself as possible (to the chassis).
How much noise you get probably depends on the quality of your inverter, but you're almost guaranteed to get some. I used a 100W inverter for a while and did get the occasional whine more than other times.
Someone can correct me if I'm wrong, but you won't really get noise on the DC line, at least not audible noise.
Running an AUX next to a power is going to increase noise.
If you're removing your indash, the existing wires should suffice for powering the screen. If not, you probably need to run new.
Have you thought about how you're going to control the XBox? Safety should be utmost importance in the car, that's why most people have a touchscreen in addition to steering wheel controls, but I don't know of any touchscreens working with the XBox. That, and the user interface should be simple and easy to use. I use XBMC on my XBox at home, but I don't think it would be friendly in the car without some serious new skinning.