I am guessing that this adaptor pluggs into the jack in your laptop where the regular AC brick would go correct? If so, then it would work just like if you were to remove your battery and plug into the wall. It will work just fine and perfectly normal. However, once the power goes out, the laptop will abruptly shut off if it is still runnning. So the battery might be good to leave in there to give it that time to gracefully shut down. Then you can even leave your computer in standby so the laptop battery is continuously drained and then recharged when the car is on, so no detremental effects to the battery and "instant-on" for music and such.
As for the hardwiring in, I suggest wiring it directly to the battery, but off of a relay tripped by an ignition wire. So the wire is only on while the car is on, and other times it is running solely off the laptop. This should negate the need for a tank circuit because the usual wall power feed to the laptop will be gone simulating unplugging it. So it will keep going until it shuts off.
You can set XP up so that when on battery power, go to standby after 5 minutes of non-use or something. Seems like a great solution.