How to prove the expressiveness of first-order logic formulas with equality over the empty signature? How can one prove that every first-order-logic formula with equality over the empty signature is equivalent to either False, or "there are exactly n elements in the domain", or "there are at least n elements in the domain"?
I assume you mean a formula with no parameters. In this case take any structure in this language = a set with no additional structure. Now look at all permutations of this structure = all permutations of the set. In this way any subset of size $n$ can be mapped to any other subset of order $n$. That is all finite sets of the same size are conjugate under automorphisms, so no formula can distinguish between one set of size $n$ and another set of size $n$. Thus the formulas you mention are the only possible ones.
I assume you mean a formula with no parameters. In this case take any structure in this language = a set with no additional structure. Now look at all permutations of this structure = all permutations of the set. In this way any subset of size $n$ can be mapped to any other subset of order $n$. That is all finite sets of the same size are conjugate under automorphisms, so no formula can distinguish between one set of size $n$ and another set of size $n$. Thus the formulas you mention are the only possible ones.