Computation indistinguishability questions
No, $X = \{X_n\}_{n \in \mathbf{N}}$ means $X = X_1, X_2, \ldots$ where each $X_i$ is a distribution.

So one could let each $X_i$ be the uniform distribution on strings of length $i$.

That means there is a polynomial $q$ such that for all $n$ and $x$, if $X_n$

assigns non-zero probability to $x$ then the length of $x$ is at most $q(i\hspace{.02 in})$.

The distributions do "take $n$ as some sort of input".

See these two questions for $1^n$.