GPS = yes, it don't really matter as your speed can always be assessed from the video footage alone if need be.
wifi are a option but it always seen haunted by ghosts so i would personally prefer a camera with a screen for the rare interactions with the camera.
Some are quite wifi addicted now, and will prefer to do all interactions with just about anything with their phone, i am not like that i plug my memory cards into my card reader on the PC and view my footage on a proper large screen.
It do seem like a few new cars can have problems with hard wire, but it is a few ones as far as i can tell by posts in here, and i assume if it is bad you can always go as far as the battery itself and connect there, but in general using a piggy back fuse in the fuse box are what most of us do.
How long you can have the camera running on the car battery depend on its size and state, but if you do deep discharge them that will cut off their expected life cycle, and of course now in winter time in a cold place the battery wil perform even worse due to low temperatures.
Also have in mind that often you have rain / dew / snow on your window while parked so then it don't really matter if the camera are recording or not.
Normally for hard wire you use a kit for it that have battery discharge protection, so you can never discharge your car battery below a voltage where you can start your car.
Also have in mind if you record a lot when parked, then you also have to drive a "lot" to charge your car battery, so if you only drive 1 hour every day and record all the other hours of the day, that will not work and it would be bad for the car battery also.
I am going to use parking guard soon, but normally i will have a cut off timer set to 30 minutes, and then once in a while i might change that timer to 2-3 hours if i park out on the town for that long ( don't happen often )
At home in my apartment i have CCTV on my car parked down below in the back yard.