Why not allow creativity of definitions? It appears to me that a fair number of issues with allowing ZFC to work with other mathematical topics is that one cannot phrase certain definitions inside ZFC. Would not this be fixed by allowing definitions to be creative? If we decided that we could define anything we want, and implicitly state that every theorem about this definition has the disclaimer "if definition exists, then...". 
I'm mainly wondering what issues crop up if you take this stance on definitions.
 A: Of course you are allowed to do whatever you want. It might just be provable that what you want is inconsistent.
More specifically, you can say something like "If $\varphi$ defines a set, then such and such and such". It just might be that $\varphi$ does not define a set, or it might depend on additional assumptions (e.g. the class of ordinals smaller than the least inaccessible cardinal might be a set or a proper class).
And in some sense this is what set theorists do. They assume more axioms, which in turn gives us more information about what sort of sets - strange as they might be - exist, and we see what sort of consequences these axioms have and how they fit together in all kind of ways.
The problem arises when you want to talk about things like "Assume the set of all sets exists, then ...", and then the assumption is always false in $\sf ZFC$, so it doesn't even make an interesting assumption. So in order to overcome these sort of problems in category theory we have universes that allow us to slightly modify the meaning of the word "set" (e.g. sets are element of this universe, and then they are elements of a larger universe). 
But even then, there's no really problem with being creative in the definitions, the question is just whether or not your creativity is worthwhile. I can imagine a world where I can walk upside down from the ceiling; but it's not this world so it's nothing more than a dream. You can dream about a world where the ordinals make a set in $\sf ZFC$ but it's not going to happen, unless $\sf ZFC$ is inconsistent.
