It’s practically impossible (-20 penalty) to hide while attacking, running or charging.
So you can remain hidden while making a spring attack, but the DC to spot you is not going to be very high. If you still manage to win the opposed roll, your target will be denied its Dexterity bonus to AC and will have an additional -2 penalty because you are effectively invisible.
From Skip Williams' column:
If a foe you cannot see hits you with
a melee attack and is adjacent to you
at the time, you know the foe's
location. For this reason, smart foes
move right after they attack; even a
foe that has made a full attack can
move after attacking by taking 5-foot
step (provided it has not already
moved during its turn).
When an unseen foe hits you with a
melee attack from more than 5 feet
away, you know the general direction
from which the attack came and that
the attack came from more than 5 feet
away, but you do not know the
attacker's location.
That is the main reason to spring attack: you need to move after attacking, even if invisible, otherwise your target will know your location. If your target has failed its spot check, you still have total concealment (50% miss chance).
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!
Best Answer
Spring Attack does not specify movement mode
Unlike Ride-by Attack and Fly-by Attack, Spring Attack is generic and works for any movement mode, including flight. It says only that you may move before and after the attack. So if your goal is to move, attack, and move again, while flying, you do not need Fly-by Attack; Spring Attack will let you do that.
On the other hand, Spring Attack is not, and does not count as, Fly-by Attack for prerequisites and the like. If your goal is to get something that requires Fly-by Attack, you have to take it separately. For that matter...
Note that Fly-by Attack and Spring Attack do not do exactly the same thing
Fly-by Attack allows you to take any standard action in the middle of your movement, while Spring Attack allows only a single attack, but Spring Attack protects you against attacks of opportunity made by the target of that attack, while Fly-by Attack does not. Generally, the Fly-by version is much better. So if you wanted to take advantage of its superior benefit, you would have to take Fly-by Attack separately from Spring Attack.
That said, just houserule this mess
I do strongly recommend a houserule here: