Parking in Japan (car tips)?

I plan to rent a car in Fukuoka because I plan to get to:

• Hita (Attack on Titan Musuem, Hita Attack on Titan dam statue)
• Kumamoto (see some of the One Piece Statues/explore area)
• Beppu
• Nanzoin Temple
• Sakurai Futamigaura

What I was concerned about is parking, generally if I need to pay for parking what's the procedure?

• Do I pay for parking at a machine, find someone to pay physically or what?
• If I park at a convenience store (e.g. Lawson's, 7 Eleven). If I buy a snack/drink, could I possibly leave my car there for up to an hour let's say, while I go explore elsewhere? Or is this frowned upon and you should only park while you're shopping there.
• If a place is private/residents only, are there any clear road signs to indicate this?
• If anyone happens to have a good guess how much my tolls would be to travel from Fukuoka to all those places that would also be great. Any good rough estimate e.g. ¥5000-¥10,000?

Any tips on renting car/driving around in Japan also appreciated, thanks! 😀