Aww, it sounds like you're going to be a great owner.
As she get older, she should ride in a car seat. That really isn't going to work for a pup, so the crate would be next best. I'd probably buckle it in if possible. I'd be sitting next to it with my fingers through the holes. lol. Or I'd get caught up in the excitement and hold her.
You may have to stop to let her use the bathroom. This could be kind of difficult though since she doesn't have all of her shots. If she is used to using a pee pad, maybe she would use one on the car floor? Really though, she will likely just hold it. Or she may go on herself in the crate. I would definitely take teh Nutrical with you.
Water should be available at all times or at least offered many, many times throughout the day...for life. It doesn't have to be available at night, but I personally would leave it out for a pup. You can free feed or feed 3-4 times a day for now (reduced to 2-3 times a day as she gets older). It really depends on the dog, your schedule, and what they do well on. If you do decide to free feed, it can be more difficult to make sure they are eating though. Obviously that isn't going to work if you feed canned food either.
Some dog like to tear up peepads and there could "possibly" be a concern with her eating them. That's a wait and see thing, but I'm pretty sure she won't. It depends how high the box is (whether she can get in and out). If it's too high for now, maybe just put the pad on the floor?