Well, that's an interesting question.

1. So this is frankly the easiest part. It wouldn't take much to design a flight suit that fits these requirements. In fact, for bonus points, they could probably put together a flight suit that would protect against supersonic flight and at sub-orbital altitudes. It wouldn't be much for shimmying down chimneys in, but like, you could get up and walk in it. Most likely flight controls would be primarily by neural interface.

2. Very doable. The stack would have to get stuffed into the sleigh during travel.

3. Now, this much is very doable. It's going to be somewhat larger than the typically-depicted "Santa sleigh", and not particularly stable in flight. But with some computer-controlled thrust-vectoring thermal jets - the same propulsion technology used in most VTOL aircraft - a flying "sleigh" could be accomplished.

4. This, on the other hand, is trickier. Reindeer definitely exist, and with some neuroelectronic surgery could even be given implants that'd make them tolerant of flight. But physically, the air velocity, low pressure, and temperature are still going to be very poor for Reindeer, and building each a flight suit with life support would probably make them too heavy. So I'm going to cheat and say, in fact, that they instead produce nine vaguely reindeer-shaped flight pods that are anchored to the sleigh by a high-tensility harness. In this way, various flight systems could be offset into the 'reindeer' modules.

5. This is actually pretty easy. Just strap a light to the front of one.

Okay, so we've got our sleigh, Santa, and reindeer (kinda). What about bonus stuff?

• Well, 'Santa' is a person, so talking to children (and eating cookies and milk!) are pretty easy.

• Getting into the house... well, if he can get out of the flight suit, it's probably doable.

• With considerable AI assistance, Santa could theoretically institute a digital analysis network that analyzes kids based on all available recorded data. It'd not be foolproof (and also viewed, in-universe, as a huge government overreach and civil rights violation). But it's doable.

This is actually a pretty big endeavor, as you're calling in some fairly high-end engineering firms for the flight modules. A lot of it is going to need to be adapted from high-end flight systems. But it is doable.

There's one big problem though: For all this effort, Santa is still bound to one world... and humanity is firmly an interstellar species in this setting. Getting Santa to go faster-than-light? That's a whole different question.