Since you're seeing all kinds of weirdness with other apps -- even though the exact details are different -- I *strongly* suspect that this is a server issue.
Re: syncronised only until 29. February 2016
This means that the messages on the server are not ordered by date (in terms UID's / sequence numbers, those are both IMAP terms, with exact definitions).
IMAP *requires* this for incoming messages, can only change if you move a lot of messages between folders, or if there is a server issue.
Even if you moved messages, any messages *received* after that should show in AquaMail.
And so I suspect a second issue too: the folder's message count is not correctly reported from the SELECT command (standard IMAP command).
Re: now the sent mails folder (containing ~1500 mails) is now not syncronised correctly
And this means that the server is not returning any messages when the app asks for "messages received since 30 days" (or whatever).
Now, Sent messages are of course not received, but they should have their "received date/time" (called INTERNALDATE in IMAP) set too, to the "when a message was sent".
So this is server issue #2 (maybe #3).
What you may want to do is: long press the account -> options and folders -> IMAP options -> change "which date/time to download" from "server time (received)" to "date header (sent)".
And then refresh the account.
Hopefully the server properly implements "search by Date header" (which Sent messages definitely do have), but I can't know if it does.
Re: Thunderbird
A desktop mail app can afford to load "everything", their sync algorithms are much more basic, they just ask the server for "all messages" and download them all.
So they (desktop mail apps) are fortunate to not be affected by any of these issues.
( but I wish Thunderbird has a setting to only sync some number of recent messages, not "everything"... alas... )