Hello, welcome to the forum. Thank you for being so wonderful and adopting this bundle of lab!
First off, remember, lots of praise for what you do want, redirect from what you don't want.
1. I would get a seatbelt for your dog to restrain her in the car. Or crate her if you can fit the crate inside of your vehicle. I have the Kurgo harness and it works fine to keep the boys in their seats. I also have the Kurgo hammock and that keeps the front seat divided from the back. Though they can climb over or under it if they really wanted to, so maybe a metal divider like what a crate is made of would be better for you. In the house you can keep her on leash until she learns her boundaries. Do you have a crate? If not, I would get one and look into how to crate train.
2. You can put a temporary gate up on your door, I did that for a while and it was very helpful for when guests or packages came. You should also teach her to sit and wait any time you are leaving. Be sure to leash her up before ever opening a door. Then you can ask for a sit, wait for the sit, then say wait (I like to combine it with a sweeping hand motion with my palm doing somewhat of a downward karate chop creating an invisible 'do not cross' line in front of my dog's face and chest). Do not let her take a step forward until you say 'okay' and allow her to do so. If she moves, repeat the process. If you want to be able to open the door without taking her out (say a guest arrives, or you are grabbing the mail etc) you could again use a gate for this situation. Put the gate up about 5 feet from the door. Teach her this is her limit, she can not cross the gate. Once mastered (she accepts the gate, settles nicely, isn't trying to bust it down), you can take the gate down and put a mat there or a line or whatever you want to mark that as her 'spot' and do not cross until invited line. It will take a lot of practice and praise but it is doable. Again if you don't feel confident and comfortable, leash her up.
3. I would guess that the humping has more to do with being under stimulated mentally and physically. Sounds like she needs more exercise and more obedience work and maybe a 'job' to do. Obviously both of these things will be tough to do while she is recovering so most of what you do will be on leash I am sure you can find something. You could easily set up some nosework training in your house. You could even work on the above training at the same time! Order some training tools on amazon, when the ups guy comes it will be a good time to teach her the 'wait' or 'stay' command that 5 or so feet behind the door. Save the boxes. For nose work you take some yummy, smelly treats and put them in one box. set out all the boxes on the floor and without indicating which box they are in, let her find the box, lots of praise and treats for finding the box, let her have the treats in the box, then take the box and put it up. Do maybe 3 or 4 runs and then put all the boxes away for next time. You want her to keep wanting more. For further training, leash her up and tether her to something study or have a second person hold her. Then get her riled up by shaking the boxes and moving them around the room. Return to her and release her to go find the box with the treats. I use the words "find it" to release my guys. You can do it on leash or off leash, up to you, but I think it is important to train it both ways. Then you make it harder, make all the boxes the same, put boxes on top of boxes, put boxes under chairs, etc. Then you move on to putting other items out instead of just boxes, then putting your treat inside of a container, like an altoid tin with a couple of holes punched in it, and then you hide the tin with a few treats on top, eventually you stop putting treats on top. It's amazing how fast they learn and how powerful their noses are. Nosework is super fun and less 'controlled' or rigid than something like obedience, but it might be something to help break up the day and give her something to do that works her brain. You can do this indoors, in your yard, in your garage, etc so it is easy to do without needed a large facility or play area.
Once she is better I would get a trainer, either get in a class or bring one to your home.
Good luck! I look forward to updates!