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 ...

Manual of the Planes (p49) says

Only a few circumstances, such as the psychic wind, an attack from an astral dreadnought, or a blow from a powerful githyanki sword, can sever this cord. Monsters, items, and circumstances cannot sever a silver cord unless this ability is specifically noted.

Planar Handbook: A Player's Guide the the Planes only mentions the ability of a psychic storm to damage a silver cord (p165), and The Eye of Gu’n’ragh magical location's (p174-5) recharge condition mentions damage to the cord for an astral traveler on a failed save.

So anything mentioning the silver cord specifically can damage or sever it, but anything not mentioned cannot. Wish or miracle might do it, but that's in the part of the spells requiring DM fiat.