Z3C is "sleepy" (no immediate notifications)

Hi there,

I recently got a Z3C and rooted it, then upgraded back to 23.4.A.0.546. I removed some apps that I dislike with Root App Deleter and installed my other favorite apps, including AdAway.

Now, the issue I experienced since the first boot is that all apps are not receiving notifications in real time, I have to open the app to "stimulate" it to fetch stuff.

So for example if my phone is in stand-by and I don't turn it on, no WhatsApp messages are received. I actually think I receive nothing until I open the app itself, and also when I am writing, it seems like the app is having troubles at getting online and delivering as it shows a waiting mini-icon on messages instead of instantly delivering them.

I am pretty sure this is not due to the rooting, maybe AdAway could be involved, but it's probably something that changed from Android 4.4 to 5.1, as I had same set of apps on 4.4 without problems.
I checked that apps are allowed background traffic, and they are. I also get this problem on either HDSPA or wifi.

Any thoughts/ideas? Thanks in advance!