Does the subset sum problem remain $NP$ complete when we increase word size and restrict number of words?
Your problem is not really well-defined, since it's not clear what is the parameter $n$. That said, under any reasonable interpretation, your problem is still NP-complete, as can be shown by padding. Given an instance of subset sum, we can create an instance of your restricted subset sum by adding a lot of zeroes to the set.

Subset sum is solvable in polynomial time if the weights are polynomially bounded. More generally, if there are $m$ integers and each one is bounded in magnitude by $B$, then subset sum can be solved in $O(mB)$. In your case the weights are not polynomially bounded.