There are two bodies, and Gate would work.
Astral Projection:
"You and up to eight willing creatures within range project your astral bodies into the Astral Plane.... Your astral from is a separate incarnation.... If a creature's original body or its astral form drops to 0 hit points, the spell ends for that creature." PHB 215 (emphasis added by me)
From the spell description we can conclude that a creature under the effect of Astral Projection has two separate corporeal forms located on two separate planes. Essentially, they are split into two separate creatures sharing a single consciousness.
Gate:
"When you cast this spell, you can speak the name of a specific creature. If that creature is on a plane other than the one you are on, the portal opens in the named creature's immediate vicinity and draws the creature through it to the nearest unoccupied space on your side of the portal." PHB 244 (emphasis added by me)
As a reminder, a creature affected by Astral Projection is both on their original plane when having cast the spell, and on the Astral Plane. Therefore, if you are casting Gate from the original plane targeting a creature affected by Astral Plane, they do count as being "on a plane other than the one you are on" as defined by the Gate spell.
What would happen is their astral body would be pulled through the Gate to your plane (a plane other than the Astral Plane). Then, as per the Astral Projection spell:
"If you enter a new plane or return to the plane you were on when casting this spell, your body and possessions are transported along the silver cord, allowing you to re-enter your body as you enter the new plane." PHB 215
As far as other spells...
Consider which body would be effected by that spell, if not both of them. Each spell would need to be considered individually and the outcome should be determined by your best judgement.
In the case of Sending, I suspect the message would reach both bodies (given the 5% chance to not reach the astral body), but would not be heard by the unconscious physical body.
In the case of Scrying, you would end up reaching the unconscious physical body should the spell succeed.
I'd say no
The way I read this:
No spell — not even wish — allows one to escape from Strahd’s domain. Astral projection, teleport, plane shift, and similar spells cast for the purpose of leaving Barovia simply fail, as do effects that banish a creature to another plane of existence. These restrictions apply to magic items and artifacts that have properties that transport or banish creatures to other planes.
it isn't like an Anti-Magic field or something where the spells themselves don't get cast, it's their effect that is disallowed, so whether the spell was cast outside of Barovia or not isn't relevant.
I also read "for the purpose of leaving Barovia" to include being summoned out of it, because the person already outside trying to "summon" the person inside still have purpose of someone leaving Barovia, which seems within the scope of what the quoted text is trying to prevent.
That's my reading of it, anyway...
Best Answer
Gate requires the name of an individual creature
The wording "specific creature" in the spell description seems clear. "Fire elemental" isn't a specific creature; it's a kind of creature. It might be a specific kind of creature, but it isn't a creature.
We know this is the intended meaning by comparing the wording to that used in other spells (emphases mine):
The description of the Locate Animals or Plants spell begins (PHB, p. 256):
The description of the Locate Creature spell says, in part (PHB, p. 256):
When the rules ask a spellcaster to choose something like "unicorn" or "cow", they use the phrase "specific kind", and the wording of locate creature explicitly contrasts that usage with "specific creature", as used in gate.