So, by Kollar-Miyaoka-Mori Thm, the Fixed Point Property holds for all Fano varieties. Another wide class is that of toric varieties. This gives a broad variety of examples.

Torsten: thank you, your comment really answers my question! Perhaps it would be better to post it as an answer, so I could mark it green...

You convinced me, except for my Question 3. Actually, in my case the bundle $E_1$ is the Frobenius push-forward of $\mathcal{O}_X$. There are easy cohomological criteria for $E_1 = \mathcal{O}_X$. What about $E_1 = F^s_\ast \mathcal{O}_X$?