For 1: a field extension $L/k$ is separable is $L$ is a separable algebra, that is, if $L\otimes K$ is a semisimple algebra for all field extensions $K/k$. In particular, if $L$ is separable over $k$ and $K$ is an extension of $k$, then $L\otimes K$ will have no nilpotent elements because it is semisimple.

For details, see for example, Pierce's beautiful book Associative algebras.