Yes, if you download to one of the Shared folders on Android 11+ no storage permissions should be required.
No, you cannot download to an aribitrary folder (like /myFolder
etc.) or the root dir of the external storage on Android 11+.
In addition,
WRITE_EXTERNAL_STORAGE
does no longer exist on Android 11+.
See also here.
So WRITE
permission should be requested this way: