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Find a common domain for the variables x, y, and z
for which the statement ∀x∀y((x = y) → ∀z((z = x) ∨
(z = y))) is true and another domain for which it is false.

Given statement is True when x, y, z $\in$ {1} and is False when x, y, z $\in$ {1, 2};

counter example (1=1) $\implies$ ( (2=1) $\lor$ (2=1) ) is False ie statement is False when x = y = 1 and z = 2.
Is the given statement true only for a singleton domain and false for every other domain?
Yes, for every domain with more than 1 element the statement will be False.

And for every singleton domain the statement will be True.