I know that everyone loves to get into the "how do you ID your repeater" debate. But as far as the OP is concerned, probably the first question that should have been asked is....
"What are you trying to accomplish with your 8900 doing cross-band repeat"???
Are you trying to setup something so you can hit another repeater, are you trying to expand your coverage area if your friends/family are camping nearby but too far away or over a hill for you to talk ht to ht.... If you want good answers on how to setup your 8900 and your HT that is the starting point.
As others have said, your 8900 is powering off because it is wired into an accessory outlet instead of directly to the battery. By doing it your current way you run the risk of getting spurious noises and background hum from the vehicle wiring, and you also can't run your 8900 without the ignition key being turned on. Most newer vehicles have a timer which allows your accessories to run i.e. fm radio, for a certain amount of time after the vehicle is turned off. Most GM vehicles will allow the radio to run after the key is turned off until you open the door, then it turns everything off. Either way this isn't ideal, and if you run your 8900 on high power it will likely burn up something in your vehicle, or burn itself up from having insufficient power.
As far as battery drain is concerned, the answer is yes, it will drain your battery. The question becomes how busy is the channel that is being repeated, and at what power level. Your 8900 set to 5 watts, running full-duplex, and with very light use might run all day in your car. Either way I wouldn't chance it being out in the woods camping on whether or not my vehicle will start. An easy way to preserve your battery is to put a voltage regulator in-line with your 8900 so that when the vehicle voltage reaches somewhere around 12 volts that the power is turned off. This should give you enough juice left in your battery to start your car. Otherwise carrying a jump-box would be a good idea. Solar power will prolong your running time but i've never seen a reasonably priced solar panel that will actually produce as much power as your 8900 can drain, so again it will prolong your use but eventually your battery will go down. You also run into the cloudy days, night-time use, etc where the solar panels are basically worthless.
Easiest way to setup your HT is to set it to 1 simplex frequency. Then program your 8900 to handle everything else. You can setup the 8900 to work on another repeater system with the right programming. And yes it will even do "odd-splits" as can be found in the owners manual. I have an 8800 and use it this way all the time with good results. As an example and for example purposes only:
My HT is programmed to 450.000tx 450.000 rx with a 79.7PL
My 8800 is programmed on the UHF side for 450.000tx 450.000 rx 79.7 PL
The VHF side is set to 145.000tx 148.000rx 79.7 PL
In my case i'm trying to hit a VHF repeater with input 145.000 79.7 and output of 148.000 79.7
Only thing to remember is you have to let your radio "drop-out" before you key-up, otherwise it will try and tx on the wrong channel, or won't tx at all. And you have to key up your HT a little longer because you are essentially keying up 2 repeaters instead of 1.
Good luck!