Does SN2 favor strong nucleophile and SN1 favor weak nucleophile? Does E2 favor a strong base, and E1 favor a weak base?
That's not the right view.

The $S_{N}2$ mechanism does not favor strong nucleophiles. That statement doesn't make sense at a high level. Mechanisms can't favor anything.

The key here is to look at what controls reactivity in $S_{N}2$ and $S_{N}1$ reactions.

The former is concerted where the bond breaking and bond forming take place at the same time. Therefore, the strength of the nucleophile is important as is the propensity of the leaving group to leave.

In the latter mechanism, there are two steps, bond breaking and bond forming, where bond breaking is the slower and more difficult step. Since the nucleophile is not involved in that first step, the strength of the nucleophile simply doesn't matter.

Similar considerations apply when looking at elimination reactions.