Flight - Yes, you must move
Specifically you have to move your minimum forward speed, which is defined in the movement rules.
If a flying creature fails to maintain its minimum forward speed, it
must land at the end of its movement. If it is too high above the
ground to land, it falls straight down, descending 150 feet in the
first round of falling. If this distance brings it to the ground, it
takes falling damage. If the fall doesn’t bring the creature to the
ground, it must spend its next turn recovering from the stall. It must
succeed on a DC 20 Reflex save to recover. Otherwise it falls another
300 feet. If it hits the ground, it takes falling damage. Otherwise,
it has another chance to recover on its next turn.
For poor and average, that is "half" of your movement speed. That means you have to use a move action to maintain that, a 5-foot step isn't far enough.
If you have Good/Perfect maneuverability, your minimum forward speed is "none", so you don't have to move.
The Hover feat also lets you stay still in the air without using a move action, so you could full round cast. You have to stop before you can use Hover in that way though, so if you want to start summoning and you're already moving, it doesn't help much.
Yes, you can cast Summon Monster
There are two ways to cast a full round spell. One of them is to use a full round action. The other one is to use two standard actions - Start and Complete Full Round Action.
The "start full-round action" standard action lets you start
undertaking a full-round action, which you can complete in the
following round by using another standard action. You can’t use this
action to start or complete a full attack, charge, run, or withdraw.
On turn 1 you use your Move action to move and stay flying, and use Start Full Round Action to start summoning. On turn 2, use Complete Full Round Action to finish summoning, then use your Move action to stay flying once more.