can babies go on shuttle without car seats? sitting on parents lap?

Yes it should be ok

Yes, as long as you have them in your lap. :)

I'm not sure? But, if it were my child I would definitely put him in a car seat because the drivers go very fast and your on main roads with lots of other vehicles!

Yes

If you're traveling alone with that child; you may not have alternative options but if you are traveling with other adults then, I recommend you drop them off at the terminal first, and theh, proceed to turn in your car to park

Probably up to you to decide that