Android's Gmail app is using Google Cloud Messaging (GCM) to push email messages/notifications and sync with the phone. I think that’s “the API” you are looking for. The bad news is that it is obviously very well protected.
You can get started for free with GCM's JSON REST API and use it for your push messaging projects, but forget about using it for your Gmail in the same fashion Google does. The only option for getting a similar efficiency would be using IMAP's IDLE extension, which uses also push.
Focusing on what you need, I think there are decent solutions for the use cases you have listed in your question… You could use a [**Google App Script**][4] or libs like [**GMail for Python**][5], which seems a valid option to me... from the [**GMail for Python GitHub**][6]:
Features
- Search emails
- Read emails
- Emails: label, archive, delete, mark as read/unread/spam, star
- Manage labels
If you are developing an Android mobile app Gmail Public Labels API could also be of your interest...
Hope it helps...
EDIT: Google just introduced its GMAIL API