Complexity of closest set of points in plane problem
There seems to be some confusion regarding parameters. There is $n$, which is the number of points we wish to return; and there is another parameter, let's say $N$, which is the number of points in the input. Assuming that $N \geq n$, the output would consist of $n$ points, so we can dispense with your $k$.

Since the heap contains at most $n$ points, all operations take $O(\log n)$ time. Therefore the running time can be upper-bounded by $O(N\log n)$. It should be easy to construct matching hard instances which show that the worst-case time complexity is indeed $\Theta(N\log n)$.

Regarding space consumption, the usual convention is to measure only the "work tape", that is, only the auxiliary space taken by the procedure, without the input and output. In this case, this is only the heap, which uses up $\Theta(n)$ space.