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):
Describe or name a specific kind of beast or plant. [...]
The description of the Locate Creature spell says, in part (PHB, p. 256):
The spell can locate a specific creature known to you, or the nearest creature of a specific kind (such as a human or a unicorn), [...]
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.
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.
Best Answer
The rules are ambiguous
Here are two possible interpretations. Since there are two, you'll have to ask your GM whether this will work:
The creature is drawn through regardless of size
Spells do what they say, and no size limitation is placed on the "drawing through" effect of the spell. Whether or not the creature could fit through the gate normally, it is drawn through by this effect.
If the spell could only draw through creatures that fit in a space equal in dimensions to the diameter of the portal, it would say so. Since it does not, this effect is not bound by the size of the creature.
Side-note: If you want a narrative explanation, most gargantuan creatures can squeeze through a space that huge creatures fit in (a condition that a 20 foot portal satisfies)
The creature can only be drawn through if it fits into the portal.
Since the spell specifies that the maximum portal size is:
Any creature that cannot fit will fail to be drawn through since you...
Determining whether a creature can fit through the portal is an issue that NautArch's answer explores in a bit more detail. It's another question you will have to ask your GM as some creatures are too large to sensibly squeeze through a 20 foot hole (even though they are classified as gargantuan).