Vehicles allow you to go to other locations faster without having to pay for a bus ride.
You improve your relationship with the characters by interacting with them and talking to them. Just don't insult them.
The facesitting scene actually has a looped animation of Jasmine smushing you with her butt. It's just that between her dark skin and her butt blotting out a lot of light, that animation could look completely black depending on your device's display settings or GPU settings.
As for your ideas and requests, I can't say or promise anything for sure.