triangulated categories - Is K(R-Mod) compactly generated when R is an artin algebra?

The answer is in general no - $K(Rtext{-}mathrm{Mod})$ can fail to be well generated even when $R$ is artinian. As you mention $K(Rtext{-}mathrm{Mod})$ is compactly generated if $R$ is of finite representation type. It turns out that the converse holds. This is a result of Jan Šťovíček which occurs as Proposition 2.6 in this paper. The precise result is:

Proposition Let $R$ be a ring. The following are equivalent:

(i) $K(Rtext{-}mathrm{Mod})$ is well generated;

(ii) $K(Rtext{-}mathrm{Mod})$ is compactly generated;

(iii) $R$ is left pure semisimple.

In particular, when $R$ is artinian this occurs precisely when $R$ has finite representation type.