Can I store an arbitrary file on an iOS device yet? What if I want to download an MP3 using Safari or Chrome and play it with the native iOS music player? Can arbitrary apps share data without specific developer support yet?
How would you share data between two apps if both developers didn't support that?
Generally speaking, if you're a developer, you can vend whatever files you want as long as you and the recipient agree on type identifiers, either copying or by sharing a destructible reference. You can't save an arbitrary file to an arbitrary location because the apps are sandboxed, as far as they can tell they're the only thing running on the phone. It's a little more careful than Android but it's meant to be part of a defense in depth.
Everything you mention is fine but I'm not sure there's some killer user story or use case that justifies it in light of the security issues. I don't think any 3rd party app developer should be able to see any of your file system ever, not on your phone. It's just too dangerous, the thing is always on the network, it knows where you live and you can't unplug it.
Visible global filesystem on a phone always seemed like a gee-whiz feature that wasn't really justified. Frankly I think the visible global filesystem on personal computers isn't really justified, considering how many people just dump everything into ~/Documents and most productivity apps have their own bespoke document browser/organizer.
Honestly I just spent about 30 minutes trying to find a website where I could even try to download an MP3, usually everything I'd want to listen to is in the podcasting app (or the site the content is on just has an app). I used to have GoodReader, which you'd launch from a URL in Safari and it'd just download anything and play it, but it sorta became unnecessary after a certain point. I'm not sure what the point of being able to play it in the "native music player" is. What app you play content with shouldn't be important, all the matters is that the desired content is available by a convenient and appropriate modality. iOS doesn't have a "native music player," it has iTunes for music in iTunes, Podcasts for Podcasts, and then Beats and Amazon Music and Spotify and everything else for their music.
One app to play all your music is 1990s thinking; modern apps are meant to brand content and service experiences, instead of them all launching the "native music player" they all call the same native sound API. The mechanics of how the media moves across the internet or across the filesystem is invisible to the user.
-- I wanna decide who lives and who dies - Crow T. Robot, MST3K
"Oh, I don't know" -- Joel Robinson.