I am trying to assign an issue to a team and that team does not appear in the list. How can I edit the team so it appears in the assigned to list? Using search doesn't find it - I just see a list of all users and other teams.

You are able to assign an issue only to the teams (and members of those teams) that own the project. To add more teams to the current issue's project, please go to the project settings > Project Teams > Add Team.