k4in

I've gotten higher than 1A ratings from a USB port (usb 3.0) before and absolutely gotten less than 1A from a similarly rated 'ac/dc module'

Can never go wrong with getting an appropriate, reliable setup. In my experience, FriendlyArm boards are no where near as fickle (power wise) than OPi boards. I can boot any of them from just about any power source I connect them to. For this reason I prefer the Neo.

OPiZ's are absolutely a different case. For me, they rarely work just from USB, unless I use a specific, good quality cable.  Forget compiling/installing more robust libraries.  I can't speak too much on the relays you mention. I can tell you though that I've purchased 4+ OPiZ boards, all serve a purpose. Require dependable power supply setup. I have a Powerstrip that genuinely pumps out 2 amps + to USB port. So I use that with a good cord and I've never had a power issue since.

Tl,DR; OPiZ over all design quality isn't the best but it's still proven to be a working product. I've purchased many of them, the only power issue I've really run into has been at fault of my own setup.

Congrats! You're one of the first thinking about non appropriate powering, so we don't have to move the thread in this subforum.   
 
The USB of my notebook delivers more than 1amp stable on USB, the reason why I never mention it in terms of SBCs is that it is above the specifications for a USB port. Not every notebook will deliver and I don't recommend it at all. 
 
1A is a bit low, normally we recommend 2A at 5V. But let's see if we get it reliable working. Cause your use case seems to be not that calculation intensive, it might be OK when you rund your OPi0 with a minimal power consumption setup. I suggest you read through this thread here:
Reduce peak consumptions seems to be important for your case (which could be problematic with wifi ). 
 
For debugging, do you have a USB-TTL dongle so that you can connect to your board in case wifi isn't on but it is powered through your 'box setup'?
 
 

I've got an OPi Zero v1.1 for a home project, It was supposed to be something very simple, connect the board to wifi, make a web app for external control and switch a power relay, I did finished it and everything was working fine but I had sudden disconnections and one day just didn't wanted to reconnect, I've read before about how this board has an horrible wifi chip and trying to avoid further headaches I went and bought an USB WiFi dongle TP-LINK_TL-WN725N, I got it today and connected it to the board using its 13 pin header, then I powered it on via micro usb and worked right out the box, no need for extra config, to finish it off I rewired it to this little box where I have it tucked away aaaand didn't work...
 
After testing why I found that the wifi would work only on the microusb power with my laptop, inside the box where it was working before I have this AC/DC 5v 1A module which power the OPi through 5v pin of the expansion header and relay (1 channel model), whenever I tried using the converter in the box the wifi didn't work, looked like it didn't  even power since there was no led activity of the dongle, but the board itself was powered and booted correctly, or at least that is what I believe because the board led would behave the same as when it would boot and connect to wifi.
 
In short, when I power it via microusb usb wifi works, when I use the 5v 1A converter mentioned above it doesn't, and for a last test I disconnected the relay from the circuit to feed just the OPi but wifi still wouldn't work, then I read this answer about usb power, if right it would mean that my laptop isn't supplying more than the ac/dc module, or is it wrong?, also I have video output and onboard wifi disabled, I really don't know what I'm missing here.