https://dev.opera.com/extensions/apis/ might not be exactly up to date.
activeTab - it's just 'tabs' API but with limited scope and we support it
gcm - partial support, that's why Join application isn't working correctly. It relies on https://developer.chrome.com/apps/instanceID which we do not support.
As for Pushbullet I did not have any problem with sending files between Chrome and Opera, so is this somehow limited for devices?