Does the Nexus 7 [2013] (Wi-Fi) tablet update to Android 6?
Yes, it is true that Nexus 7 [2013] (Wi-Fi) can be updated to official Android 6.0.1. On Google's factory images page for its devices you can see all the updates. 

The concerned apps work fine with Android 6.0.1, at least on my Nexus 6 with Android 6.0.1.