A quick, visual guide to the editions, by logo:

[first author's note: I'm throwing this up here to "seed" the process, so to say. Please feel free to add to this answer as is easy/convenient for you. Please note that imgur resizing is no longer working, there's a hack in the source of this post you can use. -n60]

[note: there is no effort made here to explain or compare editions: please see Q/A like What are the big differences between the D&D editions? or How many editions of Dungeons & Dragons are there? for such discussion.]

Logos

The "Dungeons & Dragons" logo has changed pretty distinctively among editions and is pretty-consistently used within an edition, and so provides a good starting point. (Of course there are exceptions, most currently the "special edition" covers for 5e.)

Pictured below are photos of the most common logo or two for each edition, both on the books' spines and front covers.

5e

_{The "spine" logo also appears on the lower-left part of most covers, and both these logos are in the corresponding places on the Starter Set box.}

5e also released a special edition black cover with distinctive artwork. Notably, the stylized dragon ampersand (&) in the logo is the same as the original 5e books. The black covers do not have the spine artwork on the cover. The PHB is red, the MM light blue, and the DMG dark purple.

4e

The font of 4e's logo roughly resembles the 3e/3.5e logo (see below) but without the horizontal sword image in between the two words. It also has a stylized dragon ampersand, again similar to 3e/3.5e/5e, but it does look distinctly different from the other editions (the "dragon's wing" is more prominent than in the other editions' ampersands).

3.5e, 3e

3.0/3.5 used the same logo; the logo varied a bit in execution, but is fundamentally unchanged from book to book. Pictured below are two variations, spines and covers.

Books for 3.5 will say "v3.5" both on the spine and on the cover. The (few) examples I own have the "v3.5" at the bottom of the spine and in the lower part of the cover, well away from the main logo.

Books for 3e just have the logo, no version-marker.

2e

The logo as presented in full-color (corebooks, for example) and foil. Look how nicely it proclaims itself to be 2e!

1e

Two fairly different logos; best I can surmise it's the "when" of the book's design that determines the logo used. (The newer logo establishing the now-iconic ampersand-with-dragon's-breath!) No edition number specified, since there was no need!

Various Basic

0e

Only the following elements in the core rulebooks use the Astral Plane

While it's impossible to list every possible Astral Plane interaction in D&D, the list of things in the three core rulebooks which rely on the Astral Plane is actually very limited.

The following spells, items or abilities allow travel to the Astral Plane, and will not have that function if the plane is removed:

• An 18th-level monk's Timeless Body ability
• A wild magic sorcerer's wild surge (2% chance)
• The 9th level spell astral projection
• Prismatic spray and prismatic wall's violet layer (can also banish to the Ethereal or another plane)
• The Robe of Stars
• A torn Bag of Devouring, Bag of Holding, Heward's Handy Haversack, or Portable Hole, or in some cases one of those placed inside the other
• Ether cyclones on the Ethereal Plane (5% chance)

The only other core rules elements which rely on that plane are as follows:

• Souls of slain individuals traditionally travel to their final resting place via the Astral Plane. You need to invent a new reasoning for how souls get to their afterlife or the realm of their deity.
• The Githyanki live on the Astral Plane. Without the Astral, they either don't exist or live somewhere else.
• Forbiddance blocks planar travel, including specifically from the Astral Plane (as do certain other spells blocking planar travel in general, like antimagic sphere, Mordenkainen's private sanctum and imprisonment); obviously, if you have no Astral Plane, those spells don't do that any more, but of course they still block travel to/from other planes.

You don't need an Astral Plane

Fundamentally, you can completely ignore the Astral Plane. The only significant changes you need to make are explaining how souls and planar visitors get to the realms of the gods, explaining where stuff gets banished by certain spell effects that normally send things to the Astral, resolving very high-level abilities that normally allow astral projection so that they work some other way, and explaining what happens when you place one Bags of Holding inside another.