How popular? Not in my own experience (i never saw it myself)
If we go into including all go variants (diplomatic, democratic, pyramidal, fractional,… ) because there are more as 2 players then the list is long but i don’t think this is what is referred by the word rengo. (Well not in this topic at least)
Pairgo is a name used by federation, tournament system for the specific case of male +female teams, it would be unwise to interfer.
Rengo doesn’t refer to all go variants played with more as 2 players but to a similar way of playing a game of go between 2/2 players as between 1/1. Sometimes extended to more players. The term is more a mystery for a new comer like atari or joseki but its meaning is shared by the go players community.
Team go may refer to championship organization (n players paired vs n players, or a team vs a team with direct elimination) more as what we call rengo.
Automatically yes. Now there is a very common manual practice largely used (like for tournaments for example) to calculate it and can be a nice bonus to have that automated option
Interesting
A bit too mathematical, not sure it would be understood or accepted by all as a replacement for the word rengo.
But do we really need a new word?