Yes, you can use astral projection from planes other than the Material Plane. The Manual of the Planes describes how spells that access transitive planes such as the Astral Plane function on other planes in the section "Moving Among Transitive Planes" (starting on page 45).
Characters generally must use spells or spell-like abilities to access a Transitive
Plane. ... the astral projection spell takes you to the Astral Plane ... Such spells should function in any location coexistent with or coterminous to the plane.
It includes various examples of travel between transitive planes on page 46. These examples reinforce that "Material Plane" is a placeholder and that you can use astral projection from anywhere that is connected to the astral plane. Your body will be left behind on the plane you project from.
From the Ethereal Plane to the Astral Plane: You can move to the Astral Plane from almost everywhere on the Ethereal Plane, because the Astral Plane connects to everywhere on the Ethereal Plane. ... If you're using an astral projection spell, your physical body remains in the Ethereal Plane ...
From the Plane of Shadow to the Astral Plane: From the Plane of Shadow, you can use the astral projection spell, leaving your body behind on that plane ...
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 Dungeon Master's Guide lists the Nine Hells as a single plane in Chapter 2: Creating a Multiverse (page 58). It also often mentions the home plane of devils, without ever specifying a level for any of the devils.
Mordenkainen's Tome of Foes counts the Nine Hells as a single plane. In Chapter 1 The Blood War (page 5):
(emphasis mine).
Descent into Avernus also refers to the Nine Hells as a single plane. In Chapter 3 Pervasive Evil (page 79):