So the question here is:

Do bonuses to 'your speed' apply to flying speed?

Sadly, the answer is that is that it's not clear. I'll break down the arguments for and against.

For

• The 'Using Different Speeds' section on page 190 of the PHB says that "If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speeds during your move.
• The 'Flying Movement' section on the same page says "If a flying creature is knocked prone, has its speed reduced to 0 [...] the creature falls [..]" which produces nonsensical results if 'its speed' refers to anything other than its flying speed.
• Unlike almost every other speed booster in the game, the Boots of Speed and the Boots of Striding and Springing both talk specifically about your walking speed.
• Crawford says that it does in one case, and in general.

Against

• Every other speed booster in the game just talks about 'your speed'. Some of them, like the Longstrider spell, are clearly only meant to be for walking speed.
• Mearls thinks bonuses to your speed only apply to walking speed. That said, he doesn't sound very sure, and he's not an official source of rules interpretation, so take that one with a grain of salt.
• Under the Speed section on page 182 of the PHB, it defines speed as: "Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round."
• Under the 'Your Turn' section on page 189 of the PHB, it includes this statement: "Your speed - sometimes called your walking speed - is noted on your character sheet."

To me, the last two arguments are the strongest ones, since they're the only attempts to actually define what 'your speed' means. That said, it's still a bit hazy, so I guess the answer (as ever is) has to be:

Talk to your DM. It's up to them to decide whether this works or not. I would definitely point out to your DM that this boost of 10 feet is effectively all you'd be getting from the Mobile feat, because:

• The second benefit of Mobile is pointless to you, since you're going to be flying all the time anyway.
• The third benefit of Mobile is equally pointless to you, since you're a flying Sorcerer. There's very little reason you'd ever get into melee, and even less reason for you to make melee attacks. (Although this works with melee spell attacks, so you can, for example, Vampiric Touch and then fly away without provoking opportunity attacks.)

Feats are meant to be powerful, and if all you're going to be getting is an increase from 60ft -> 70ft, it's not really a big deal. In fact, if I were you I probably wouldn't take Mobile. Haste is trickier, because it's a very small investment that would boost your speed up to 180ft (with the bonus Dash action), making you faster than any creature in the Monster Manual. Of course, it doesn't make much sense to allow one and not the other.

Yes, it's halved

Mobile: [...] Your speed increases by 10 feet.

And

Prone: [..] Standing up [from prone] takes more effort; doing so costs an amount of movement equal to half your speed. For example, if your speed is 30 feet, you must spend 15 feet of movement to stand up.

So mobile increases your speed, and prone costs you half your speed. If you have 30 speed and add 10 from mobile you get 40, if you stand up from prone that turn you have 20 left since standing up costed you 20. Nothing contradicts this general ruling, so it's RAW and (most likely) RAI.

If you have a hard time wrapping your head around this think it this way: mobile allows you to run faster, but not to move faster. So when you are standing up you still consume half your movement to stand up, since you can't stand up faster than someone without mobile, but once you stand up and start running you still can run faster than someone without the feat since your running speed is higher, just not as much distance as normally since you only have half the time to run.

However, as @PixelMaster said, you could get the Athlete feat to reduce the cost of standing up from prone to only 5 feet of movement, thus benefitting from the full 10 speed of the Mobile feat.