I am a plus user, all I do is hit the refresh icon (bottom left) when in the inbox on the iphone. Any new emails will show up. I do not use email push on Iphone however.
as far as calendar, you must switch be a calendar beta user (you cannot switch back to the old after) then click on the following link to adjust your calendar settings. Its all I use and it works wonderfully. Add calendar entries on your Iphone or the webpage and they update each other seamlessly.