1 - Android's system settings app has a switch to turn off all background sync. Apps are supposed to always observe it (i.e. not sync anything in the background when this is off), but Aqua gives you flexibility.
I've had too many cases of "this app doesn't work" -- to discover that the setting, which defaults to "allow background sync" was turned off, presumably by the all-too-helpful sales people in Verizon stores...
2 - It's 25 most recent messages -- to keep up to date, every time the app checks mail.
Up to date means that the app would check that they still exist, remove any deleted ones, and update the existing ones to their current state (i.e. read / unread, starred state for IMAP).
3 - Custom check interval means something not in the selection list (which has 15 minutes, 30 minutes, 1 hour, etc). So if you need to have the app check mail every 14 minutes -- not 15, not 10 -- then this is your setting.
Reference time point -- explained in the FAQ.
4 - It means that folders that you accessed "as needed" (as opposed to folders set to sync every time -- like the Inbox does) will be removed from the account list and the navigation drawer after one week.
5 - No, it does not download all messages.
It downloads and keeps up to date a limited number of messages per folder (2 is the default), and stores on the device, but does not always show, up to 250 (so you can scroll down a message list and see those older messages faster).
None of this (how much is kept up to date, or cached) delete anything from the server.
