I have a company email,
firstname.lastname@example.org, which is managed by my company admin and it is hosted on Google Apps mail services.
I want to know whether the admin can check my emails or my chat history.
If your organization uses Google Vault, an admin can search your email and chat archive.
Also remember that as an admin, I could always reset the user's password, log in as that user, and access everything.
Unless you have used some sort of encryption, you should always assume that anything you email can be read by systems-admin type people for any server that your message enters: this includes the admin at your company, the admin at the receipent's company, and (because of the way the internet works) any number of people in the middle. Essentially, email is like a postcard, visible to everyone along the way unless you take steps to protect it.
I don't know about chat - but in general I'd assume that anything put into a computer can be recorded and probably is.