Let $R$ be ring such that every left $R$-module has finite projective dimension ( resp. finite injective dimension). Is the left global dimension of $R$ finite?
Similarly, Let $R$ be ring such that every left $R$-module has finite flat dimension. Is the weak global dimension of $R$ finite?
Yes. If $R$ has infinite left global dimension, then for every $n$ there is a left module $M_n$ with projective dimension greater than $n$. But then $\bigoplus_{n\in\mathbb{N}}M_n$ has infinite projective dimension.
The same argument works for weak dimension.