From what I've read over the years a subwoofer is a subwoofer and car subs can be integrated into hometheater. Basic design principals would need to be followed. Enclosure size, etc. But don't ask me about them as I don't have a clue. Seek the advice of a subwoofer expert.
Car stereo speakers don't fair as well in HT since they are designed for a completely different environment. Look at how and where the speakers are installed in a car as opposed to a home speaker. The fact that they are "separates" is much better but the results probably wouldn't be that great. I know since I've tried it in the past on a few occasions. Cheap HT speakers still sounded better. HT separates can sound great in the car though.
My advice would be to sell the Kickers and use the funds to get some proper HT drivers or HT speakers. You can try it if you would like. You would only be out some wood and some time. But be prepared to be disappointed.
Using a power converter with a car stereo amp to power them is beyond my experience but it doesn't sound like a good idea. I'd use a proper 120 volt power supply (receiver, separate amp).
I just picked up a pair of Polk LCi65 in-walls ($75/ebay) which I'm going to dis-assemble and put into cabinets some day (year) and I expect the results to be excellent (providing I choose the correct size internal cabinet volume) since the drivers, tweeters and crossovers were designed for the HT environment. Or, if I can find another pair on the cheap I might slap them in my ole Jaguar and see how they sound.
But iIn-walls and speaker kits are a great way to get started in speaker building.
"Everyday room": Panasonic 58" Plasma, Dish HD DVR, Pioneer Elite vsx-23, BDP-23 BR, dv58avi universal dvd player, Dynaudio Audience 50's, 40's 122C, Dayton HSU-10 subwoofer.
"Movie/Music room": Toshiba 65" DLP, Dish HD receiver, Adcom GF-700 5.1 receiver, OPPO BDP-80 BR player, Polk LSI25's-LSi7's-LSiC, 2 original Dayton 10" "Mighty-Mites" subwoofers. (subject to change without notice).
Also have...too much to list. Help me.