Familiar is described a few different ways in the PHB, but the context seems generally the same down the board. I will provide a few (not all) quotes if I can find them.
Clairvoyance: "...in a location familiar to you (A place you have visited or seen before)..." -PHB pg 222
The entry for Scrying on PHB pg. 273 breaks down knowledge of a place or creature pretty well. Familiar: You know the target well
It boils down to there not being a written definition of what is or is not 'familiar' to your character. This means you may have to deliberate with your DM if you are uncertain. Are you familiar with your adventuring group? More than likely. Are you familiar with the Orc who carried you over the cliff into the ocean a few weeks ago? You might think so. Are you familiar with the necromancer you've only heard stories about and can pinpoint its exact location? No.
If I would have to make a ruling about familiarity, I would refer to the Scrying entry.
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
You don't.
Generally speaking, the rules are silent on whether spellcasters know details about spell failure. However, in this case, I think that the wording specifies that the caster cannot know. The spell text draws a distinction between the sending and the arrival of the message; the caster is involved only in the actual sending of the message and has no control over its arrival.