The default 5E cosmology is unclear on some of the connections. I'll give the best answers I can give here on how the Astral borders other planes in the default cosmology:
The prime Material Plane (also, is the Prime Material Plane exactly the same as the Material Plane?)
The Prime Material Plane is how older editions of D&D referred to what 5E calls just the Material Plane.
Update due to release of Spelljammer setting: As of the release of the Spelljammer setting, 5E explicitly describes a physical mechanism to reach the Astral from the Material, namely, that once you reach the edge of a local system's Wildspace (the vacuum-filled region within what 2E referred to as the system's crystal sphere, outside each planet's atmosphere), you move from the Material to the Astral (this is a significant change from 2E, where you transitioned to the Phlogiston, a purely Material location separate from the Astral). Thus, in 5E, if you have a Spelljamming helm and a ship to install it in, you can travel physically from most campaign settings to the Astral Plane and back (Eberron and Athas being exceptions; Eberron is described as being largely isolated from the Great Wheel, while Athas, at least in prior editions, had a sealed crystal sphere and therefore could not be entered or left via spelljamming).
No other direct connections from the Material to the Astral are described (though it does indicate color pools can form from the Astral to the Material), and the Astral lacks a border region the way the Ethereal does (it is "next to" the various Wildspace systems, not overlapping them).
The Astral is described as the realm of thought and dream, so it's metaphysically "close" to the Prime Material. Oddly, color pools can form in 5E from the Astral to the Ethereal (but not Inner Planes), indicating a level of connection that didn't exist in older editions' cosmologies.
The material echo planes (the Shadowfell and the Feywild)
Every reference to the echo planes I've been able to find considers them equally connected, or disconnected, from other planes as the Material. Since the Astral is accessible from the Material plane, it's presumably equally accessible from the Shadowfell and Feywild, and "borders" them (or not) equally.
The Elemental Chaos
The Astral naturally connects to many planes via color pools, and said pools can never lead to the Inner Planes, so on the most basic level, no, the Astral doesn't "border" the Elemental Chaos (or any Inner Plane) the way it "borders" the Material Plane; if you want to go from the Astral to the Inner Planes, you need a spell to skip intermediate planes, natural color pool connections don't form.
Aside from that, it's ugly. 5E doesn't describe how the Inner Planes (which the Elemental Chaos is part of) and the Outer Planes (which are connected to each other and the Material Plane by the Astral) connect, if they do at all. In 2E, where Planescape nailed things down in great detail, the planes were rather like a stack, with the Outer Planes on top, the Astral between them and the Prime Material, the Ethereal between the Prime Material and the Inner Planes (rather like the World Axis alternative from the 5E DMG, but without the Elemental Chaos and with the Abyss as the CE Outer Plane). The Astral connected directly only to the Outer Planes and the Prime Material, the Ethereal only to the Prime Material and the Inner Planes (the Ethereal also housed the demiplanes). There was even an explicit idea of planar "distance" that affected how clerical magic functioned when you were more or less distant from your god (and reinforced the whole concept of Outer Planes being on the opposite side of the Prime Material from the Inner Planes).
But in 5E, the details in the text are vague and the illustration of the planar cosmology (PHB page 303) could be interpreted to have the Astral surrounding the Inner Planes, which in turn surround the Material plane. It describes no such connections in the text, perhaps indicating the relationship isn't simple enough to be depicted in a drawing. The DMG is the only place they describe the relationship at all, and it's incredibly (probably intentionally) vague:
The Inner Planes form a wheel around the Material Plane, enveloped in the Ethereal Plane. Then the Outer Planes form another wheel around and behind (or above or below) that one, arranged according to alignment, with the Outlands linking them all.
Note the text I emphasized, indicating the wishy-washy positional relationship between Inner and Outer Planes.
This doesn't even mention the Ethereal or Astral. The Astral remains largely associated with the Outer Planes, and the Ethereal with the Inner, but it's essentially DM fiat whether there is any sort of (meta)physical ties from the Elemental Chaos to the Astral.
