Some of these are browser limitations, such as AFAIK you can't download anywhere other than the downloads folder. Local file management is one of the things we'll be improving with the desktop builds, where it can be on par with C2.
Subfolders are on our todo list.
Cloud save should just save directly back to the cloud without opening the file picker dialog if you click 'Save' a second time - please file a bug if that's not happening!
Saving to local browser is safe as long as you click "Request persistent storage" in the About dialog. Obviously this ought to be available in that dialog as well, but once you've gotten persistent storage permission it won't automatically delete anything, unless you explicitly tell Chrome to. I agree we need to make some changes to make this clearer and increase people's confidence in using it.