Does this mandate allow me to use a. 3rd party Teams, Google Chat and Slack client?

I suspect the answer is no, but why?

There are criteria for how dominant a platform is to be considered gatekeeper. Teams, Google and Slack have much smaller market share for private messaging, so I guess they are not affected. Don't remember the criteria by heart.

They are not designated gatekeepers. It is unfortunate because interoperability should be its own objective for its own sake.

Because they're not big enough to be considered a "gatekeeper".