I'm not familiar with the definition of when a Lie subalgebra is reductive in a Lie algebra. Please can you remind us of it?

In his textbook, Humphreys sets as an exercise that a finite-dimensional representation of a reductive Lie algebra $L$ is completely reducible if every element in the centre of $L$ acts as a semisimple endomorphism.

Does "semisimple" for representations mean "completely reducible"? If so this is false as the only Lie algebras for which all finite-dimensional representations are completely reducible are the semisimple Lie algebras.

Alexey, is that really the way you spell your surname?

One shows, by fair means or foul, that an automorphism of $\mathbb{Q}_p$ must be continuous. For instance it suffices to prove that an automorphism preserves $\mathbb{Z}_p$.

The collection of Borel sets on $\mathbb{R}$ has this cardinality, but not the collection of Borel sets on an arbitrary locally compact $X$.

Isn't it the Cayley graph (under tbg's defintion) for $S=\emptyset$?

The class number is irrelevant. If $K$ is a number field, then $K^*$ is isomorphic to the direct product of a finite cyclic group (whose order is the number of roots of unity in $K$) with a free abelian group of infinite countale rank.