Operations on semi-hereditary rings

I am learning about left (right) semi-hereditary rings, and got stuck with the following questions. Just to recall that being left semi-hereditary means that every finitely generated submodule of a projective left module is projective.

Let $S$ and $T$ be two unital non-commutative rings which are left semi-hereditary.

(1) Is it true that the opposite ring $S^{op}$ is left semi-hereditary. I expect the answer to be no, but my knowledge of non-commutative ring theory is not enough to get a counterexample.

(2) Is it true that $S\otimes_{\mathbb Z} T$ is left semi-hereditary?

The answer to (2) is "no" even for hereditary rings. For example, if $S=T$ is the algebra of upper triangular $2\times 2$ matrices (or, more generally, pretty much any finite dimensional hereditary algebras) over the field $\mathbb{F}_p$ ($p$ prime), then $S\otimes_\mathbb{Z}T$ is not hereditary (and since it's a finite dimensional algebra over a field, not semihereditary).