No, for full flying
The Change Appearance portion of the alter self spell says (emphasis mine):
Change Appearance. You transform your appearance. You decide what you look like, including your height, weight, facial features, sound of your voice, hair length, coloration, and distinguishing characteristics, if any. You can make yourself appear as a member of another race, though none of your statistics change. You also can't appear as a creature of a different size than you, and your basic shape stays the same; if you're bipedal, you can't use this spell to become quadrupedal, for instance. At any time for the duration of the spell, you can use your action to change your appearance in this way again.
The important part to note is that it explicitly states that none of your statistics change, and you retain your basic shape. Adding wings that would be large enough to allow you to fly is a major alteration to your basic shape (humanoid). I think the portion that covers being quadrupedal sufficiently demonstrates that a major modification in methods of travel is outright denied. Especially since if you can't change your arms into legs (which is a very minor change considering they function similarly), logically you wouldn't be able to change them into wings either. The intent of the spell seems to be for small changes such as:
Skin pigmentation, reshaping ears, eye colour, hair colour, talons, claws, teeth, and other minor modifications. The most major modification is definitely the ability to breath water, but even that simply allows you to extract oxygen from a medium other than air, which is another minor modification.
How I would rule as DM:
I would personally say no to wings, but yes to a bat-like membrane (picture a wing suit) between your legs, arms and body. I would even house rule that you could get half speed flying out of it, and of course, gliding. Since you don't have hollow bones, or a small mass like a bat, that's why I would personally impose the half speed limit. This conforms to the restrictions of the spell and would permit some creative experimentation.
No, it doesnt.
The Fly skill page says it requires no action, the check should be taken as part of another action or reaction to a situation:
Action
None. A Fly check doesn’t require an action; it is made as part of another action or as a reaction to a situation.
You will notice that a hover check is required when you ignore the minimum movement required to remain flying, which includes not moving (or hovering):
Without making a check, a flying creature can remain flying at the end of its turn so long as it moves a distance greater than half its speed. It can also turn up to 45 degrees by sacrificing 5 feet of movement, can rise at half speed at an angle of 45 degrees, and can descend at any angle at normal speed. Note that these restrictions only apply to movement taken during your current turn. At the beginning of the next turn, you can move in a different direction than you did the previous turn without making a check. Taking any action that violates these rules requires a Fly check.
Note that monsters have a specific feat on the bestiary to allow them to hover without making skill checks. This means that even creatures with a natural (or magical) fly speed still need to make Fly checks, even to make a full-round attack or take a 5-foot step while flying.
Best Answer
While a creature's paralyzed, it can't opt to fly from place to place even if its flight is magical and wingless
The Pathfinder Core Rulebook FAQ includes this exchange:
(Emphasis, link, and editorializing mine.) You can read a multpage thread that predates this 2015 ruling that debates this topic here.
Were the GM to make a house rule saying that the spell fly and other methods of magical wingless flight do allow a creature to jet from place to place by taking purely mental actions, the campaign would have to deal with the subsequent fallout from such a ruling. That is, any wingless creature that the DM rules flies by mental effort alone—and that'd probably be all the DM's applicable monsters—can now fly away despite being affected by a hold person effect, for example. For this GM, that would be bridge too far, and, while this player wouldn't abandon a campaign in which such a house rule was made, this player would urge the GM reconsider as—seriously—casters are already powerful enough! If unable to get the GM to reconsider, this player would find for his PC a means of such flight posthaste.