Meta data like To From MessageId InReplyTo are unencrypted (only transport encrypted - not end to end).
The Subject and Contents of a message are fully end to end encrypted when the padlock is visible.
Yes, DeltaChat on android has even an button in the settings that clears the space on the provider,
This doesn’t remove them from your chats, as the messages, once fetched are stored in the local database on your device.
Yes it’s on our roadmap:
See List of Feature Proposals for more feature Proposals and requests
That’s the reason why it’s in the Background - so if you’re device doesn’t kill the background process you should receive push notifications.