It's impossible to prove whether God exists via empirical evidence, i.e. via observation. It is not impossible via logical proof. Logical proof is proof of a higher order.
Would logical proof be able to tell us "what God wants", such as rules, laws and stuff we should or shouldn't do?
Not logical proof for the existence of God, no. But logical proof of a Universal utility function for something along the lines of self-actualization, yes.