The "Options" menu (the gear symbol in the upper left corner) lets you disable push notifications - some or all. I have them all turned off. You can also turn off character animations there if they bother you.
I've been told to view the Leaderboard and that will sync my phone's data with server, but that does not always work for me. On multiple occasions I've checked the leaderboard then immediately Shielded, and 10-30 seconds later I'll receive a notification that I've lost a match or three, and now my score is 50-100 points less than what it was when I Shielded.