Logical statements can be proven true/false. Definitions are not logical statements, they do not have truth values, therefore cannot be proven neither true, nor false. These are mathematical logic basics.
Yes. However, in some cases (though probably not the ones relevant here) a definition can be proven to be incoherent (or, to presuppose something false), which is vaguely similar to “being false”.
It would be difficult for a definition to make any presuppositions. You could have a definition that defines some set in which a contradiction is involved ("an integer is special if it is both prime and divisible by 4"), but then you'd say that the set so defined is empty, not that the definition is incoherent.
It is quite common for a lemma to be needed to ensure that a definition is well-defined. The term “defi-lemma” exists for a reason.
As a simple example, suppose X is a set and r is a relation on X. If I define Y := X/r , the set of equivalence relations with respect to r, this implicitly assumes that the relation r is an equivalence relation.
