Mail works even worse with Gmail and Google apps after upgrade

I had good hopes that Apple would solve the mail issues with the 10.9.2 update, but unfortunately it seems that now the synchronization between Gmail/Google apps is completely off.

After archiving a message in my mail application the message remains in my inbox in Google apps and Gmail.

When I was using 10.9.1 this was not the case, I only suffered delay in receiving mails and could not manually move a message from my inbox to the all messages box.

I have Outlook on my Mac, which came with Office, but never configured it.

Do I really need to start to use that programm?