Loote,
There are different ways to represent F-stat if you did not know about it. Check your books. If they are missing it, then the author of that book is missing it. I used the R2  to make it easy for people to understand because most people know that R2 increases with increase in the number of variables whether significant or not. Just try to play with the formula if you want a proof of it.

As I said before, the answer is not definitive. It can go either way depending on the explanatory power of the variable, whether it is significant or not.  As long as the explanatory power is able to offset the effects of loss in degrees of freedom, F-stat should go up. By the way, what do you mean there is nothing to do with explanatory power? What does F-stat measure?

Oh man....even though I am not an expert in this area, I work in this field. Come on!!!