Great suggesions. We have been putting some thought into both of the use cases you suggest.
Your first idea need not be a feature of Stash alone. I've been considering whether this would be something that should be also possible from outside of your Stash folder. That is, if it's backed up, why shouldn't you be able to share a file or folder? The main objection I can come up with is that backup does not happen as immediately as updates to your Stash, so it might be a little harder to control. But to me, that seems more like something to change about how backup works. In any case, it's something we're seriously considering.
On your second idea, you're saying that you'd like the ability to declare sub-folders in your Stash, and share those with a customer, partner, colleague, etc. It's a read-write use case, compared to the read-only use case of your first suggestion. Correct? If so, then I agree that authentication for doing this is an absolute must. Similar to your first suggestion, this is part of the plan that we're currently building out. It's very unlikely that you'll see it in this private beta, but Stash is definitely the foundation for creating exactly such a feature.
I'd like to throw my vote behind this as well. I use Mozy for all of my online backup. I have a case in which I need to share a set of medial files with an SRS flash card program (Anki) that keeps its decks in sync with the mobile version of the app running on my device. To share the media files with the flash card program, I need them available on a public share.
I'd prefer either to put these files in Stash, or simply share out the latest version of these files from my Mozy backup. Currently, there's no way to do that, and I have to put the files on Google Drive instead. I'd prefer to keep everything in Mozy.
Re: Sharing out backup files - if Mozy moves to a model where any file or folder in Mozy can be marked for automatic sync, then we get the best of both worlds. Personally, I'd be fine just having the ability to make Stash files public.