Yes. If your child is in a moving vehicle, she should be in a car seat with the restraint straps in place. In automobiles (including taxis), she should be in the rear seat and using the vehicle restraint system. The car seat must be used in busses also, although vehicle restraint systems are rare on busses.
Yes. It is recommended that whenever you transport your baby that they be in a proper fitting car seat. You can definitely still get an accident in a bus or taxi and your baby unrestrained is at great risk for injury.
Yes. It isn't legally mandated in most jurisdictions, but it would be a smart thing to do. I have been in a taxi collision once and i would not have wanted an unrestrained baby to have that experience.
Yes. In buses, it's possible to hold your child, though a bus with a seat belt and an installed car seat is best. For a trip through town, most won't install a seat, but will carry the baby, and while not ideal, is ok for short trips. In a taxi, yes, car seat laws still apply, and there must be a car seat. Some taxis may have one if you call ahead, but most require you to bring yours.
No. The wording on the california highway patrol's "ca child restraints law" poster from january 2005 is not completely clear. It does say that busses are excluded from the law, as well as the back seats of taxis (as long as the passengers are fare-paying riders). The implication is that if a parent owns the taxi and is driving his baby around, the baby would have to be in a carseat in the back.