Can you attack in the middle of your mount's movement? No.

This was clarified by Jeremy Crawford in his twitter feed a couple of years ago:

Q: rider on controlled mount wants to attack mid-move. Do rider and mount share one turn, or does rider need to Ready attack?

Jeremy Crawford: A rider and a controlled mount have separate turns, but they have the same initiative, which means you decide which one goes first.

So you can't freely act in the middle of your mount's movement; it isn't your turn.

You could ready an action to make your attack when you get close enough, but that adds some limitations:

• Triggering a ready action is a reaction, so you can't do this and then, say, make an opportunity attack of your own, or cast shield when the guy you just hit decides to retaliate.

• Readying a spell requires concentration, even if the spell you're holding normally wouldn't. This puts a pretty significant extra limitation on you as a spellcaster: You can't ready a touch spell attack while maintaining a concentration spell at the same time (which of course includes all the good buffs and debuffs).

• You can't use a bonus action while you're near the target, because your bonus action has to happen on your turn rather than your mount's. Some classes/items/feats/etc and dual-wielding can give you various 'attack as a bonus action' abilities, which couldn't be used during a "ride-by stabbing" like this. A sorcerer's quickened spell metamagic pops to mind as a relevant example.

• If your class lets you make multiple attacks with an attack action, while afoot you can break up your movement in between individual attack rolls (Player's Handbook p.190); but you can't do that while mounted. If your attack action is a ready action, your triggered action interrupts the mount's movement rather than happening concurrently. (Or to put it another way, because it isn't your turn, you don't have any movement to break up.) And if you stop your mount next to a target and attack during your own turn, you can't really move without getting off your horse first.

Does Flyby protect you from opportunity attacks? Yes.

Here are the relevant rules concepts:

1. When you are mounted, your mount moves and you are taken with it; you aren't using your own movement.

   [Your mount] moves as you direct it, and it has only three action options; Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it. - Player's Handbook, p.198

2. You only draw opportunity attacks when you use your own movement to leave an enemy's reach.

   You can make an opportunity attack when a hostile creature that you can see moves out of your reach... You also don't provoke an opportunity attack ... when someone or something moves you without using your movement... - Player's Handbook, p.195

Based on rules 1 and 2, you do not draw opportunity attacks while mounted; only your mount does. But...

3. A special rule relating to Mounted Combat says they can try to hit you anyway.

   If the mount provokes an opportunity attack while you're on it, the attacker can target you or the mount. - Player's Handbook, p.198

That's the key to understanding how mounts and opportunity attacks work. If your mount has an ability (such as Flyby) or uses an action (such as Disengage) that prevents opportunity attacks against it, that also inherently prevents opportunity attacks against you. You don't provoke on your own, and your mount isn't provoking any attacks that could be redirected to you.

Any mount can use Disengage, but Flyby is better because it allows you to use Dodge or Dash at the same time as not provoking attacks.