Being in a coop should allow you to control the island a bit more like being able to /lock /unlock could be helpful, i use to be in a coop and i always wish i could open and lock the island when the leader wasn't on so that was a pain. A command like /coop promote <playername> would be a good command.
Things like /lock and /unlock yes. However maybe not things like /ni replace or /coop kick, only the leader should control them things .
Yes that would be a good idea like the leader can do like /moderator [Player] or something like that.
Or maybe /coleader? Dunno, I think it makes more sense than moderator, which might lead to confusion, like "was your co-op member a moderator?" versus "was your co-op member a co-leader?". You should add it to the main suggestion thread, either way. A command to give a coop member these permissions would be great. Support.
I support this, under the condition they don't get commands like /coop kick or /trust because who knows how badly they'll be abused?