Does the solvent play a role in determining which pathway the reaction will take, SN1 versus SN2? How?
It affects more their reaction rates.
The solvents plays a minimal role in determining the mechanism as
Polar protic solvents can hydrogen bond (or form a strong intermolecular ion-dipole interaction) with the nucleophile and therefore, weakening its nucleophilicity, however, polar aprotic solvents do not interact with the nucleophile.
This is because,
Here is a video that explains in details how to predict the mechanism as