Can cut the end off of a cigarette plug adapter that has three hole dc sockets and wire it to a fuse or to the battiery.
yes, you can. be sure to put a 15 amp fuse in the hot wire, and ground it good to the body and you will be fine.
Which do you think would be Better the battery or the fuse?
where you pull power from is up to you. if you go from a fuse, you could pick one the cuts off when you turn off the car, so you don't drain the battery. going straight off the battery would allow you to power items when the car is turned off. but you need to put a separate fuse on the power side going to the outlet whether you come from a fuse or the battery to protect both the vehicle and whatever you are going to plug in.