This is actually a really interesting inconsistency.

According to d20SRD.org...

Each character within range of a gaze attack must attempt a saving throw (which can be a Fortitude or Will save) each round at the beginning of his turn.

You could interpret this to mean that the monster with the higher initiative actually has the disadvantage! Since they rolled a higher initiative, they have to roll or divert their eyes first, which implies they may either die or be not-looking come their enemy's turn, which would affect their gaze attack.

Under these circumstances, I think one has to remember that different turns during the same round are supposed to happen concurrently-ish. I think the emphasis isn't on "at the beginning of his turn" as much as it is on "each round". Making the roll "at the beginning of his turn" seems more like an organizational issue - everyone rolling at the start of the round or on the gazer's turn would slow the game down.

As such, even if the first monster dies or turns to stone or diverts their eyes, the other monster should still have to - at the beginning of the round - roll in some way to 'resist' the other's gaze attack.

Example

A beholder and a medusa walk into a bar and roll initiative. The medusa rolls higher and goes first; at the start of its turn it rolls against the beholder's gaze attack, fails and horrifically turns into skittles.

The beholder still has to roll against the medusa's gaze attack; rolls, fails and turns into funk. Combat is over.

The bartender says something witty about the medusa being ugly and beauty being in the eyes of the beholder. WOKA WOKA WOKA!

While the general section on beholders in Lords of Madness doesn't give any particulars of the furniture in the beholder's lair, there is an example lair detailed later in the chapter (page 56). I won't reproduce the whole thing here, but the important room for our purposes is the beholder's personal chamber:

This large chamber is quite impressive; the walls and ceiling are covered with intricate carvings depicting beholders eating humanoids, and vast fields of mushrooms with writhing tendrils. On the ceiling above, the image of a single immense beholder watches everything. Two large, uneven stalagmites protrude from the floor along the northeastern wall, and to the south stand several well-made statues of powerful-looking adventurers and fearsome monsters. In the center of the room sit numerous desks, each of which is covered with mounds of papers and books.

This chamber is Sekarvu’s personal lair, and where it spends the majority of its time.

There is no mention of a bed, or indeed any furniture beyond the desks. So either the beholder sleeps on a desk (face it, we all have occasionally), or it continues to levitate while it sleeps.

A possible source of further details on beholders would be the AD&D 2e book I, Tyrant, which was the book in the Monstrous Arcana series dedicated purely to beholders.