is there any way to sync my mailbox via IMAP? I could just download new messages, but the old ones (and existing folder structure) was not synchronized.
IMAP4 in doogiePIM doesn't sync email over a live continuous connection, it only downloads the messages at the given command. This is to enable your other email clients, such as Smartphone, to also obtain the messages.
IMAP4 open sync with a continuous connection may be a possibility in a future version, as long as many other users wishes it to be that way.