The browser is important to be able to easily open PDFs without having to figure out which App Store app lets you wirelessly sync and open them and it lets you sort those miscellaneous files where you want. It's also important for uploading to websites - say you want to insert a .gif into a forum post but can't hotlink it from the site it's on, you download it, put it on tinypic.com and insert it in the forum post.
It seems to me that for most users "putting files where you want" is just a hassle, especially if the relevant files are available to the relevant app. If all the PDFs I have on an iPad present themselves to whatever is being used as a PDF reader, why should I care where they "are"?
As far as stuff like downloading/uploading, Apple is using typical user scenario hooks to get around exposing file system structure. In your case, Safari gives you the option to download an image, and the image view app gives you the option to "share" an image to a given URL, both via the agency of in place modal "popovers" of the sort Apple is using for the new form factor.
Now, I'm not saying either of those strategies (app specific file database, modal in-app file moving) is ideal for every user. But it represents a new, appliance like way of making the vast majority of "computing" tasks dead simple for the average user, and I think it's a mistake to dismiss this strategy just because it fails to act like a hierarchical Finder. It's different, even radically different, and intended for a different relationship to computing-- not a broken version of anything.
The streaming music in background case is legit, and almost certainly will be addressed at some point. The others, I'm not so sure. Especially with the greater hardware leeway, I would suspect that "app switching" (in which an app has its state saved, is closed, and another app is opened with a read of its saved state, if any) is close to instantaneous. Given that on a screen this size the one app at a time presentational scheme makes imminent sense, how different is that really from actual multi-tasking? Unless I want to use a device where I drag around tiny little windows, I'm only interacting with once app at a time, anyway (and I have to note here the sort of hilarious use cases that people keep coming up with, in which they are apparently surfing, working on a file and sorting their music literally [I]simultaneously[/I).]
I haven't played that particular game but isn't the developer best practice to save state on quit? If it's starting the level over after being switched away from, it isn't very well written.