I am just adding a tiny bit that is missed from the previous 2 answers, that is, the resp. part of Question 1:

> If $F^l$ are sheaves on $X$, is $\oplus_l F^l$, which exists in the category of presheaves on $X$, automatically a sheaf on $X$?

The answer is **NO**. For a counterexample see [this answer][1].


  [1]: https://math.stackexchange.com/questions/2498298/products-and-sums-of-sheaves