I think I got all of that. (One exception: here, as I'm sure you know and as Mark just explained to me, you can also log in via FB or google.)
(Also: my alias email address joined a group last night via invitation, did NOT log into Groups.io, and was able to see everything in that group. This could be a bug, though.)
I still think that the home page "apply for membership in this group" button is misleading, since it requires a password. Once I get the other answers from Mark, I can create a wishlist for how I'd ideally like it to be.
Meanwhile, I think I've come up with a workaround for recommendations to the group from outsiders. The recommendation can still recommend my old Yahoo group and provide a link. When the person clicks on the link, they would see my (home grown) instructions for joining my Groups.io. group. But ideally, I won't have to do that. Ideally, Mark will make it very easy to subscribe to a group from the group's home page, without being side-tracked by the "apply for membership" box and having to provide a password.