If you did not already, start by having a look to the Deutsch's algorithm, which is the single-qubit version of the Deutsch-Jozsa algorithm you are looking at.
Here you have some quick resources:
It is simpler than Deutsch-Jozsa and a good stepping stone.
Regarding your first question, there are $n$ inputs to $f$ and the task is to determine if $f$ is constant or balanced (under the promise that it is one or the other); you can do this on a quantum computer with a single query to $f$. If $n=1$, then classically you can do as you suggest, input $0$ for one query, and input $1$ for another query, for a total ...