So i have the following statements:
I can prove 2 and 4 are true, but i struggle to decide whether 1 and 3 are true and false. I would say neither of them are true because thats the whole point to add the seaf property, right? But im not sure at all especially about 1.
Common counter-example to 1 and 3: On $X=\{a,b\}$, we can declare a presheaf of rings (to be able to speak of units in the first place) by letting $\mathcal F(X)=\Bbb Z$, $\mathcal F(\{a\})=\mathcal F(\{b\})=\Bbb Z/2\Bbb Z$ (which also makes $\mathcal F_a=\mathcal F_b=\Bbb Z/2\Bbb Z$), and the projections as restrictions. Then the global non-zero section $2$ is $0$ in each stalk, and the global non-unit section $3$ is a unit in each stalk.