SMS only fetched when I turn the screen on
That might happen because Android 6 has a function called Doze.  Try going into Settings - Battery and look for Battery Optimization. Add there the apps that you want to make sure they sync / ignore battery optimization and see if it still happens. 

http://www.androidcentral.com/inside-marshmallow-what-doze-how-do-i-use-it-and-what-does-it-do