Q2) You mean $X=BG$, right? Then the answer is yes by standard homological algebra.

Q1) The answer to the first question is no, e.g. $X=M(\mathbb Q,n)=S^n_{\mathbb Q}$ the rational $n$-sphere has homological dimension $n$ but cohomological dimension $n+1$ (since $\mathbb Q$ is flat but not projective as an abelian group). For $X$ finite I don't know the answer in general, but if $X$ is in addition simply connected then yes, since the chain complex of $X$ is quasi-isomorphic to its homology and the projective and flat dimensions coincide on finitely generated abelian groups.

Q3) <strike>This is true if $X$ is simply connected (Whitehead's theorem) but false otherwise, e.g. $X=BG$ with $G$ acyclic.</strike> Was already answered in the question.