Final Fantasy 15 – Is Manual Driving Faster than Ignis Driving?

There is no option to wait until specific time of day. You can camp/use hotel anytime, to wait until next day morning (6:00 AM). Also, when you take hunting job, where monsters appear only at night, game will suggest you to skip time until night starts (8:00 PM). Those are the only means to skip time I've found in game.

I'm not really sure about opposite situation with monsters - when you take job for monsters, which appear only on day, during night.

Update: please note, that while progressing through game, actual time of morning/night is changing (morning starts later, night starts earlier), so times specified above, are correct only for first few chapters.

It depends on your upgrades. When you first get the car, you have a limit of 3 miles before you have to refill. Various upgrades increase this range (car wax for better mileage, gas tank upgrade to hold more gas, etc.).

After completing the game (spoilers),

you can get an upgrade that negates the need for fuel.