Let $X$ be some space, I have basically 2 questions:

1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on paracompact spaces are acyclic, but a lot of people require for paracompact spaces to be Hausdorff too, so I don't know if this works on non-Hausdorff paracompact spaces, and...

2 - In case soft sheaves are acyclic on non-Hausdorff paracompact spaces. Why don't I define the trivial topology on $X$? (Spaces with the trivial topology are paracompact) That way aren't all the sections in the sheaf trivially global sections? I know it's kind of dumb but wouldn't that automatically make the sheaves soft?