There currently is no way to change the extension chat badge.
This is currently not available for extension chat clients
No they are not partners, they just have a verified account with the badge added to it (Partner get that badge by default, but its not exclusive to them). And yes they have the moderator badge because the broadcaster modded them on the channel and they can use the channels emotes bcs they are subs on that channel. either by someone gifting it to them or the broadcaster using on of their free subs for the bot
That is another limitation to the extension chat client iirc. emotes dont rly work (not sure on that one)