A 626No. You can take a move action. One of the things you may do with this move action is dive, but you can do other things if you want to, including going back up (subject to whatever limitations your maneuverability allows, of course)

A627Short answer: no. Longer answer: even if you had rideby attack, you would still not technically be able to do this by RAW.

Mounted combat is really messed up and doesn't work by strict RAW. This is kind of a complicated question to answer. Theoretically, rider and mount have their own sets of actions (and separate initiative counts, for that matter) and act separately from one another. That means if your mount's init is higher than the rider's, the mount would move and attack or whatever else it does and then when it got around to the rider's turn, they would have a round's worth of actions to do stuff with.

In practice, the mount is usually said to move and attack (or whatever) on the rider's turn and the rider can attack concurrently. If a mount used flyby attack, the rider couldn't attack during the flyby attack unless they readied an action or similar because you're not normally allowed to butt in during someone else's init unless you're making an aoo or something. Basically riders move mounts freely within the limits of their speed on the riders' turn.