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If you know that 1/2 * base * height = the area, then the second diagonal can be seen as the height of the triangle cut by the first diagonal. Since b = 2 h,

1/2 b h = 1/2 ==> b h = 1 ==> 1/2 b b = 1 ==> b b = 2 ==> b = sqrt(2).

I can't read cuneiform so I don't know if it makes this argument.

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You'd first have to prove that the diagonals bisect each other (so that you can say b = 2h). I'm not sure whether that result was known at the time, but it's not too hard to prove.

You're right. That's pushing it for those handful of scratches on the tablet. Who knows, maybe they had some lemma tablets lying around.

From my own limited experience, ancient documents tended to be very elliptical; when everything had to be copied out by hand the reader was expected to do a little more work to understand what documents meant. So this tablet may have been the equivalent of notes accompanying a lecture: "As you can see from the triangles on this diagram..."

The tablet just gives the lengths of the sides and diagonals. There is no sign of any explicit reasoning on it. The author of the pages linked to from here thinks that the maker's decision to show both diagonals of the square indicates that it's meant to illustrate a proof. I am not convinced.

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