I too have recently included using this chip in a project but have yet to get it to work consistently.
I find the data read back from this chip to bounce around (unreasonably) by times or to vary significantly from what I get from a DMM. Even though I am using an isolated power supply to power the chip/microprocessor I am wondering if there is some isolated ground/ground loop problems. Don't know.
If I understand the data sheet correctly the vrms, irms and active power values (registers 0x20, 0x21) are the values calculated during the last half cycle. Try looking at the averaged values. If you set register 0x0C to some reasonable value (mine is currently set to 0x820 giving avg1=32 and avg2=16) try looking at the vrms and power average values (register 0x26/0x28 for short term average, avg1, and 0x27/0x29 for avg2). Note you will need to set the chip into "Customer Mode" first to write to register 0C ... and power cycle it after changing the value.
ZC count working even when no power connected? Seems very strange. I'll have to check what happens on my board. Haven't looked at that.