Can a Tag Group for custom types be added to this menu? The All Tag Types folder is a hard place to find things.

All of your custom types are grouped in the containers folder

That works. Didn't think of these as containers but it's obvious now that you mentioned that.

I would like to create my own Tag group to place some of my custom tags into.
Specifically to access only my tags in a dropdown list select.
Can I create a new Tag Group and add my custom tags to it"?

i am pretty sure you can create your own custom CONTAINER as well.