Yes, to be ample amounts to be a generator of the derived category. Of course you have to consider the sheaf together with its shifts.

Yes, to be ample amounts to be a generator of the derived category taking into account $\otimes$-powers, specifically an invertible sheaf $\mathcal{L}$ is ample if and only if the family $$ \{\mathcal{L}^{\otimes t}[n]\,/\, n\in \mathbb{Z}, t \geq 0\} $$ generartes de derived category of quasi-coherent sheaves on the scheme. This is explained in greater detail in Neeman's JAMS 1996 "The Grothendieck Duality Theorem via Bousfield’s Techniques and Brown Representability" in the general setting of divisorial schemes.

I guess this answers question 1 at least.