*1) When the car starts, power is cut to the cigarette lighters. This forces my CD player to lose power as well, meaning I have to not only turn the player back on, I have to start from the beginning of the album. I think I could solve this on a laptop by leaving the battery in, so that it'll stay booted.*
Well - that is the easy way out - but I'm sure your labtop will not like any type of short power spike over repeated times. Honestly - from me working on vehicles for 4+ years - DO NOT COUNT ON THE CIG. LIGHTER. First off it is ment to get the lighter hot. It is not ment to carry current for long periods of time. The wires that they run back there are not very big (I'd like to say under 12 gague) and I personally do not feel safe running even a CD car kit off of the cig. I have had vehicles come back with burned wires for this. And you want to run a labtop off of this?! I say run a dedicated wire from the battery to behind/underneath the dash. You can use a relay off of the ignition wire so when your vehicle is off - everything will be off (or forced to run off of battery power) but double check this - I am not too good with Honda's wiring details. Also - just look at a wiring diagram. Lots of times they use a harness for other vehicles - so you may be able to get a un-used live wire. W body cars on GM (Impala, Montie carlo, ect...) are quite nice for these!
*2) How will I be able to control the computer, especially starting the music? That's quite a distance between the spare tire well and my bum--and I don't want to run a wire between the two, since that's the thing I'm trying to avoid whenever I can.*
I don't know of any way becides running any wires. If you labtop has an S-Video out you can just run one wire under the carpet/seats for ease of choice. I'm sure that If you look around outside that people have had this problem- and i'm sure some one/company has made a program that has a CD/Track disk changer made so you can just velcro/place the changer up in front and just run one long cord to your labtop? (Maybe even wireless?)
There is a TON of sound stuff for vehicles. Just Google it and it's gonna take some time and i'm gonna say you are gonna have to do some wiring...
If you have anymore car tech questions let me know. I can get you help/specs/whatever b/c I work at a dealership.