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://mathoverflow.net/questions/212548/supplementary-notes-to-mumfords-the-red-book-of-varieties-and-schemes/212551#212551