For a discussion of "Sheaf cohomology wih compact supports over any topological space" you may want to look up "Sheaf theory - Breadon" and " Sheaves in topology-A.Dimca". It should be noted that: Etale cohomology with compact support requires the existence of proper embedding (one shows that result is independent of a chosen embedding), so it is not a straightforward generalization of "Sheaf cohomology wih compact supports" to the etale site.

A minor typo : $k$ is missing in the first line.

You can look at the book of "Neron Models" - BLR, for a more detailed description in somewhat more generality.

Of course, how stupid of me!

Thanks, but I am not assuming that $char(K) = 0 $ so this argument needs to modified (in an obvious way) to the case of supersingular elliptic curves?