Yes. Take $\mathcal L$ the trivial line bundle, with a one-dimensional space of global sections, and $\mathcal M$ a nontrivial torsion line bundle, so $\mathcal M^k=n$.

Then $\Gamma(\mathcal M)$ is certainly zero-dimensional, since otherwise $\mathcal M$ would have a nonvanishing section and be trivial or a somewhere vanishing section and then $\mathcal L$ would have a somewhere vanishing section and be nontrivial.

This provides a counterexample. Such examples exist on any smooth proper variety with a nontrivial Picard variety, meaning $\operatorname{dim}_kH^1(X,\mathcal O_X)>0$, such as curves of positive genus.

Yes. Take $\mathcal L$ the trivial line bundle, with a one-dimensional space of global sections, and $\mathcal M$ a nontrivial torsion line bundle, so $\mathcal M^k=\mathcal L$.

Then $\Gamma(\mathcal M)$ is certainly zero-dimensional, since otherwise $\mathcal M$ would have a nonvanishing section and be trivial or a somewhere vanishing section and then $\mathcal L$ would have a somewhere vanishing section and be nontrivial.

This provides a counterexample. Such examples exist on any smooth proper variety with a nontrivial Picard variety, meaning $\operatorname{dim}_kH^1(X,\mathcal O_X)>0$, such as curves of positive genus.