I've only been using AquaMail Pro for a couple of days now, but intermittently I'm getting duplicate entries for an incoming Gmail message. My Hotmail account is also configured, but I have notifications disabled for that account.
For your Gmail account, which folders (aka Gmail labels) are configured to be synced in Aquamail?
If you included "Inbox" and "All messages" (by default, only "Inbox" is included), then the same message appears in both, and hence you'd be getting the two notifications.
If this is the case, I'd recommend to have only one of those folders to be synced.
Also, when I get the duplicate or multiple entries in the notification bar and click the notification to launch AquaMail, the app opens on the "Accounts" screen as opposed to the Smart Folder screen, even though "Settings > Look and Feel > Start the app in = Smart Folder." If I launch the app via the AquaMail icon on the home screen, it opens on the Smart Folder screen per the app setting.
If I get a single message notification, i.e., no duplicates or multiples notifications, and click the notification, the app launches and opens the new message, which appears to be the logical behavior.
When you tap on the notification, Aquamail opens as closely to the message(s), for which the notification is, as possible.
So, one message -> that message is opened.
Several messages in one folder -> that folder is opened (message list)
Several messages in different folders -> account view with those folders
I am not using Smart Folder myself, so, I am not sure if it is possible to configure opening Aquamail from notifications in the Smart Folder or not. I am guessing, it might not be possible.
The option that you mentioned is for when you tap on Aquamail thumbnail, not on the notification.
On the other hand, if I get a single message notification and clear the notification by swiping, I get another notification after approximately 1 minute. Do I need to expand the notification and select "Mark read" to stop the repeating notifications?
Notify about = New unread messages
Just in case: check if this setting is check-marked: Settings -> Message Notifications -> Reset on notification clear.
If the duplicated notifications are as I proposed above, due to the same messages in two folders, - then it is quite likely that this behavior is also for the same reason. It's just the instance in one of the folders gets delivered slightly later, after Aquamail has checked that folder. Hence the delayed notification.