Wiring light switch with neutral (Z-wave)
Too bad I haven't updated answer when I rewired everything - but better late, then never, right? Here are the answers to my questions:


White wire is neutral in US in most cases... so it was in my case.
Basically, two whites were neutrals for two different circuits. I bought Voltage detector and separated right white wire by turning off circuit for lights - the other white still had electricity running through it so I was able to figure it which white/neutral I needed
Yeah, I just cut it, add in neutral connecting from the switch and then put cap on
For this you need Voltage Meter, there are quite few tutorials on YouTube that explain how to figure out which wire is which with Voltage Meter.