Subset with modified condition, is it still NP-complete?
What precisely are the problems? I may be missing something (and cannot do comments yet). Are they

(1) Given a set $A \subseteq \mathbb{Z}$ of $n$ elements, does there exist some subset $S \subseteq A$ with $\sum_{x \in S} x \le M$.

(2) Given a set $ \subseteq \mathbb{Z}$ of $n$ elements, does there exist some subset $S \subseteq A$ with $\sum_{x \in S} x \ge M$.

If so these problems seem clearly in $\mathbf{P}$, basically by the reasoning you described -- we just want the minimum/maximum possible subset sum, and then we compare that with $M$. Assuming empty $S$ is allowed:


for (1), add up all the negative numbers in $A$ and see if it's $\le M$. If Yes, then return yes. If not, return No.
for (2), add up all the positive numbers in $A$ and see if it's $\ge M$.


Again I may be missing something, but it seems like the other answer's reduction might not be addressing the possibility that $SS_\le$ and $SS_\ge$ would return yes based on different sets? Like consider input $A = \{1, 3\}$ and $M=2$.