Not a faster lock, air pressure is actually more accurate on the z axis for the most part. GPS is not great at height, and including an extremely accurate barometer is a big help.
Because the SATs only give you a pseudorange distance between you and the sat, so each say is most accurate solving for distance to/from that sat, and much less helpful resolving angle to the sat.
With a clear SkyView, around just under half of the sats are hidden by the earth.
This means that you get a full 360 degrees of data that can be near the horizon helping resolve lat/long.
But only about half of that sky is helpful for altitude. The birds you can't hear directly below the earth would be the most helpful for improving the altitude fix if you could hear them.
Baro is handy because you can take the absolute altitude from GPS as a low frequency baseline and use the baro for high frequency changes. Then when GPS says we teleported +200 feet when a new sat comes into view, we can temper that that with baro information.
I know nothing about this, but I imagine two people at the same latitude and longitude could be at different altitudes.
A trivial example would be people on different floors of a skyscraper—although I suppose gps works poorly indoors anyway. Still, even outdoors there are peaks and crevices, and on a steep slope a very trivial change in lat/lon could lead to a major change in altitude.
I would imagine it's because your distance to the satellites changes more when you move along the ground than when you move up and down the same amount.
Sure, assuming you are not in a building, balloon, helicopter, plane, drone, etc. Not to mention interesting topology like a cliff where an error in long/last of even a single foot could mean your altitude would vary by 1000 feet.
One problem is that using position to get altitude would require a detailed elevation map and that would use a lot of storage space or require internet.
But elevation maps are not detailed enough and position is not accurate enough to get accurate elevation. Think about standing on trail along steep slope. The position not being that accurate is fine since you know you are on trail. But altitude could vary wildly going up or down slope, or even up or down trail. It is probably similar to GPS vertical accuracy, but were going for more accuracy that barometer provides.
The maps are only used when resolving a 3 bird fix. As soon as you lock a fourth bird you can solve 3D directly rather than leveraging an onboard spheroid. Many receivers refer to a three bird fix as a 2D fix for that reason.
It's one factor in estimating altitude you feed into an extended kalman filter. If the GPS altitude is holding steady but there's a sudden jump in pressure that's probably a storm front not indicative of motion.
