[RPG] Dragonfire Adept – Humanoid Shape invocation

Is Flyby Attack basically a midair version of Spring Attack, or does it actually grant a second move action while airborne if I attack in that round? Does it stack with Spring Attack?

Flyby attack lets you make any standard action in the middle of a move, so it's much more versatile than Spring Attack. However, it doesn't defend against AoO! This lends itself to drastically different usage than Spring Attack, despite the superficial similarity; flyby attack works best with ranged/magic attacks, such as a breath weapon.

The text of the feat specifically forbids a second move action.

Would Hover and Wingover be useful, or does Draconic Flight already confer their effects?

Draconic Flight grants good maneuverability, which already gives the ability to hover (the feat hover has an error in the description). Wingover probably wouldn't be that useful by itself. However, if you really want to improve your flight capabilities, you'd be better off taking Improved Flight (Races of the Wild) to improve your maneuverability class. RAW this isn't allowed, since it says it requires "natural flight", but I think anyone who can has an intrinsic ability to fly meets the spirit of the requirement. So you could try convincing your DM of the same if this is important to you!

What are some feats that can help me make the best of my 3 natural weapons (2 claws, tail)

Multiattack, certainly! I imagine there are some dragon specific feats that might help, too. But, I'm not super familiar with Draconic Adept, so you might want to make sure focusing on your natural weapons will be effective. Glancing over the class, it has a poor BAB and no melee abilities, which makes me doubtful that your claws+tail will be very useful if you stick with the adept!

Only with a House Rule That Permits It

Because the dragonfire adept supernatural ability breath weapon says that "you gain a breath weapon that you can use at will as a standard action," and Metabreath Feats in the Draconomicon says, "To take a metabreath feat, a creature must have a breath weapon whose time between breaths is expressed in rounds" (66), the answer is no, a dragonfire adept can't take metabreath feats.

Although there's an argument that the time between a dragonfire adept's breath weapons is 0 rounds--or even 1 round using the breath effect ability--, that duration is never explicitly expressed that way, instead using this language: "Also, you can't apply the same breath effect to your breath weapon in two consecutive rounds (though you still can use your normal fire breath weapon every round)."

Seriously, only the DM knows how an effect functions in a game wherein the DM permits abilities to be employed by creatures that specifically can't employ them.

A Somewhat Murky Workaround
This might not be the real question, though. If the real question is, "Is there a way for a dragonfire adept take metabreath feats despite his breath weapon's lack of a recharge rate measured in rounds?" then I point you to general feat Power Surge (Dragon #313 57) and janusjones's Dragonfire Adept Handbook, which discusses how to do exactly that and many more dragonfire adept features in much greater detail.