Are these three subset sum problems hard? How to solve them?
Your work is mostly correct:


SSP1 is clearly in NP. Moreover, it is NP-hard by reduction from SUBSET-SUM: given an instance of SUBSET-SUM, reduce it to the corresponding instance of SSP1 with $k = n$.
SSP2: Let $m$ be the number of negative elements in $\{a_1,\ldots,a_n\}$, and let $S$ be the sum of the $\min(m,k)$ smallest of them. Check whether $S \leq \sigma$. This is a polynomial time algorithm.
SSP3: Can be reduced to SSP2 by negating the set and $\sigma$. (We can also solve it directly just like SSP2.)