But you are aware that the IRFZ44N has a threshold range of 2V min to 4V max right? That means you might pick one that would not switch at all at 3.3V! Also, you've really got to spend time studying the datasheet. Those thresholds are quoted for a Drain current of a whopping 0.00025 Amps
What you need to do is a plot for the "transfer characteristic". This will show how the Drain current varies with Gate voltage. Here's the plot for the part you've chosen:
Does it still look suitable now? Sorry to be the bearer of bad news
scropion86 wrote:btw any problem to use GPIO16 as output to trigger the relay as per the last schematic.
and no plan to use any power saving mode
If not using wake from deep sleep then GPIO16 is OK to use as an output. What's odd about the pull-up you got from the Arduino link is that GPIO16 has on-chip pull down (the only pin to have this) and at reset, the GPIO defaults to output anyway. With an external pull-up as shown, setting the pin to an input would make it float somewhere in the middle!