All Questions

Let $K$ be a (locally compact) local field and $G$ be a linear algebraic $K$-group. Does the topological group $G(K)$ have a cocompact solvable closed subgroup? If $\mathrm{char}(K)=0$, it is true ...

Is every connected solvable algebraic group a Borel subgroup of a reductive group? If a counterexample exists, I would ideally like it to be over $\Bbb C$.