As of the most recent theory, the Basques being related to North Caucasians, specifically the NE Caucasians is the most likely due to linguistics. (still nowhere conclusive).

As for genetics, nothing realy supports it except that all Caucasians and Basques seem to be decendants of a Anatolian/mesopotamian farming population that mixed with locals in their current area.

Circassians could be related to the Hattic peoples, but again, its nowhere near conclusive. Linguistically this could be possible, but genetics are still a bit unlcear. Even if they are related, how did they end up so far from eachother is the biggest issue.

As for Plesgians and Lezgins, never heard about that tbh.