Do Coaches become Former Coaches when the "Team" has been completed?

Not necessarily. Axiom manages the Coach role differently than other roles in the system. Where a lot of things are tied to the status of the class being completed or active, coaches are tied to the broader school year.

Anyone tied to a "team" as either a coach or assistant coach has that role until the school year itself changes. The system stops automatically giving them the "Coach" role on the team detail record if you remove their permissions from any teams they are assigned to or change the role from "Coach" to "<None>" or "Other". (This applies to the role on the Coaches tab of a team record only).