What is the best way to set up Google mail, contacts and calendar? (consolidated)
The Verge gave a good writeup recommending the Gmail App + CalDAV + CardDAV: http://www.theverge.com/2012/12/5/3732364/best-way-gmail-google-calendar-iphone-how-to

The Gmail app solves both downsides you mention above. For CardDAV, it does a 15 min fetch, which is plenty good enough for me, so I don't see lack of push as being a real problem.