You can park your car in visitor center parking. You have to take shuttle to go inside the park. Sometimes its difficult to get parking in visitor center. Try to get there early in the morning. You can also park in springdale and take shuttle to get to the park.
Of you show up really early in the morning or later in the afternoon, you have a good chance of being able to park inside. If I remember correctly no one really drives around because you cannot see much from the car. There are 2 maybe 3 trails that have very limited parking, like at most 8 spots for parking. Other then that it's easier to go to each trail on the free tram they have.
No. Travel past the visitors center is via free park shuttle, assuming you even can get parking near the visitors center. Many visitors park in the nearby town of Springdale and take a shuttle into the park from there.
Thanks! Your answer is awaiting moderation.