Firstly, cute dog!

My doggo is currently not allowed in my car, but we have this cover for my girlfriend's car and it stays put very well. The underside has little rubber grippy dots that hold well to the leather seats, there are elastic loops that go around both the front and rear headrests, elastic on the corners that you can use to secure the corners to the rear seat uppers, and extra long straps that velcro which would allow for securing along the bottom of a seat if it was possible. It's not possible in her car, but we manage to tuck some of the excess fabric in between the seat and the sides of the car and it holds tight. Also, there are some padded flaps that are tucked into the rear seats between the bottom and the upper that very securely hold the cover in place. It honestly hardly moves once it's installed.

For door panels, we purchased these but haven't technically used them yet. I imagine it would work fine, and I know it will fit in the FoST (tested it). But it would require applying a little velcro to the door panel if you wanted to get a more secure fit.

As for biting the seat cover, we haven't had that problem with our pup because we spent a lot of time training her on what she is and is not allowed to chew right when we got her (I read a lot about this and was worried she was gonna destroy everything so I was extra prepared, lol). I would recommend throwing a chew toy in the seat with her so she will have something that she enjoys and can chew on while she's in the car. If she's not searching for something to chew on, then you will be able to worry less about her chewing on the seat cover.

I can't comment on how to keep the fur from getting everywhere, though, and this is the main reason why I haven't let my pup ride in my car yet. She sheds like crazy and fur flies everywhere when we open the windows while driving.