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.

Yes, it applies to all movements

Unarmored Movement (PHB, 78) States:

Starting at 2nd level, your speed increases by 10 feet while you are not wearing armor or wielding a shield.

It does NOT qualify as your Walking Speed (or Swimming Speed, or Flying Speed)...just Speed. This applies to all types of movements and would add additional speed to the Fly spell as well.

This is supported by Jeremy Crawford as well.

Unarmored Movement is intended to increase a monk's innate speed, including an aarakocra monk's flying speed.

He further clarifies Bonuses and Penalties that apply to Speed in general:

Bonuses/penalties to speed apply to your speeds in general, unless the text specifies walking, flying, etc

As the Unarmored Movement does not specify a specific type of speed, it applies to all.

Fly Spell specifically

Fly states (PHB, 243)

The target gains a flying speed of 60 feet for the duration.

The implication here is that you (the target) had no Fly Speed (or a lower Fly Speed) before casting and it is now a speed that you, the target have. Once it becomes your speed, it becomes subject to Unarmored Movement.

Similar to Special Types of Movement in the PHB, page 182)

While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed.

Here, Swimming has a penalty, unless they have a Swimming Speed. Once they have that Speed, from whatever source, then Unarmored Movement should apply.

Rule of Cool

And it's fun! Monks are supposed to be super quick. Let them be super quick in all their speed!