The answer to the first question is almost always no, see <em>Roos, Jan-Erik(S-STOC)
Derived functors of inverse limits revisited. (English summary)
J. London Math. Soc. (2) 73 (2006), no. 1, 65--83. </em>.

<b>Addendum</b>: The crucial point is that infinite products are not exact. The most precise counterexample statement is Cor 1.11 combined with Prop 1.6 which identifies the stalks of the higher derived functors of the product with what you are interested in. Formally, it doesn't give a counter example for a single $X$ but Cor 1.11 shows that for any paracompact space with positive cohomological dimension there is some open subset for which your question has a negative answer. It seems clear that one could examples for specific $X$.