Maybe it should be emphasized that these questions deal with finite dimensional Lie algebras over a field of characteristic 0; in prime characteristic there are further complications. BR has addressed the first question and observed that a subalgebra of a reductive Lie algebra containing a given Borel subalgebra (in other words a parabolic subalgebra) will have the same type of abelian Cartan subalgebras found in the big algebra.
The follow-up question is more problematic, since the notion of "semisimple" element in a Lie algebra isn't always intrinsically defined. In fact, there is an intrinsic Jordan decomposition precisely when the Lie algebra equals its derived algebra (which for instance excludes nonzero abelian Lie algebras): this is proved by building on the classical theorem for semisimple Lie algebras, in a paper by Cagliero and Szechtman here. (The intrinsic nature of the Jordan decomposition involves good behavior under linear representations. Leaving that aside, parabolic subalgebras of a given semisimple Lie algebra do contain "semisimple" and "nilpotent" elements relative to the big algebra but not relative to their own representations taken in isolation.)
Anyway, I'm not sure right away how much can be said about Cartan subalgebras in a non-reductive Lie algebra which equals its derived algebra. Examples?