So my question is the title, but I also have a question about something else. If you have a problem, how can you determine the reason that it is in NP. So for example: given a directed graph with $N$ capacities along the edges, and a number $k$, is the maximum flow in $N$ larger than $k$?
Why is the above problem in NP?
If $M$ is a nondeterministic Turing machine accepting $L$ in polynomial time $p(|s|)$ for $s \in L$, then in the course of the $k p(|s|)$ steps it takes to accept $s$, it can only use at most $p(|s|)$ many cells of $M$'s tape.
I'll briefly address your second question, though really you should post it separately. You show that a problem $A$ is in NP by reducing it to a problem $B$ that's already known to be in NP, using a reduction procedure $f$ (function) that's at most polynomial time. A reduction maps instances of $A$ to instances of problem $B$. In order to conclude that $A$ is also in $NP$, clearly $f$ has to be at worst polynomial time, otherwise it wouldn't follow that $A$ is in $NP$.