I would like to sync contacts and calendars with Outlook 2010 using the Z-Push activesync implementation and caldav/carddav/imap.
Does Outlook 2010 actually support ActiveSync or how does it connect with exchange-servers?
No version of Outlook supports ActiveSync to your own Exchange Server. Outlook 2013 has ActiveSync support but it is not supported for connecting to your Exchange Server, only "outlook.com".