I'm pretty sure that plenty of those kind of questions are covered in Cassels' book.

The modern approach to this kind of problems follows from dynamics on homogeneous spaces via Dani's correspondence, and in-particular this issue is tightly related to the question of divergence under the geodesic flow (or more generally, one-param. diagonalizable subgroup) of certain pieces of unipotent flows.

The main tool is the so-called quantitative non-divergence theorem of Kleinbock-Margulis (generalizing previous qualitative works of Margulis, Dani-Margulis) - "FLOWS ON HOMOGENEOUS SPACES AND
DIOPHANTINE APPROXIMATION ON MANIFOLDS" https://arxiv.org/pdf/math/9810036.pdf

In particular your question easily follows from their extremely general Theorem A.

Edit - just to make the answer clear, the answer for 1 is Yes, and 2 is dealt in the Kelinbock-Margulis article (and many further developments, say by Kleinbock, Shah and others).