Peter,
There are three ways how messages can be synced:
1. Automated sync initiated by Aquamail:
(a) Scheduled background sync (where you set up mail check interval)
and optionally
(b) When certain type of connectivity appears (e.g. Wi-Fi. - there are a few options that configure that.)
2. PUSH (aka "IDLE") - where the connection is maintained open with the server (this uses more battery and data, but delivers message immediately after it arrives to the server.)
3. Manual check (you can click on the "rotating wheel" icon for checking all accounts, individual account, or, when you are in the message-view list, pull down at the top - to sync the particular folder.
Please refer to the FAQ:
* How can I specify my own mail check interval?
Most of the questions you are asking are already covered in that FAQ. So, take some time and read it, you'll find plenty of useful information.