Does input space size contribute at all to the runtime of an algorithm?
We measure the running time as a function of the length of the input, not as a function of $n$.  So, it doesn't matter whether the length of the input is $2^n$ or not.

When we talk about an exponential-time function, we normally mean exponential in the length of the input, not exponential in some other parameter (like $n$).

For your problem, the length of the input could be $2^n$, but it will typically be a lot less than $2^n$, because typically we'll have many fewer sets than that.  Either way, it doesn't matter, because we're measuring the running time as a function of the input length.

Typically, the format/representation of the input doesn't affect the running time much; the running time will be identical for any reasonable way to represent or encode the input.  So, typically that doesn't matter.  In the cases where it does matter, be careful to specify the input representation, or ask if it isn't specified.