Looking at the rules for mounted combat (PHB p. 198) and flying (PHB p. 191), you should bear the following in mind:
- The griffon acts and moves on the player's turn, and can only take the Dash, Disengage and Dodge actions.
- If the griffon falls prone or has its speed reduced to 0, it stops flying and falls to the ground.
- If the griffon is forced to move, or if something knocks the rider prone, the rider must make a DC10 dexterity save to stay mounted.
- If the griffon provokes an opportunity attack, the rider can be attacked instead.
Attacking the rider of a mount doesn't have any special penalty. Given all that, the RAW answers to your questions are:
- The griffon can move as much or as little as it's rider wants on each of it's riders turns. If the griffon Dashes, it can move a total of 160 feet in a turn.
- A creature can take the ready action with the intent to strike the griffon or rider as soon as they are in range. Note that if the griffon is moving out of range without Disengaging, it will provoke an opportunity attack that can be made against the rider. Any spells or abilities that reduce the griffon's speed to 0 or knock it prone will cause it (and the rider) to fall to the ground, so they are also good options.
- A character riding a mount doesn't gain any defensive advantage. There is no penalty for attacking the rider of a mount, assuming the rider is also within the attacker's reach.
Things do get much worse for the creatures on the ground if the rider has the Mounted Combatant feat, of course.
You can also set up encounters where the enemies have cover, or can retreat into spaces too confined for the griffon to fly or enter. On the other hand, if the player enjoys having their character fight from griffin-back, then you shouldn't take that away from them. You can also add some flying enemies to their encounters.