Why not get Ethernet hats and do POE with a switch? That might put you over your expected costs but might be easier to deal with.
yes, it will work - i have done exactly this (wired a pi to the 5v rail on a PSU) with no issues, but i used a 5v buck converter to regulate the voltage as i didnt trust the PSU fully. should be fine without a voltage regulator too though. but i would still rather use a 5v project power supply, you can get a 5v 20a power supply for $20 on amazon and wire every pi to it
The pi will pull the current it needs, the power supply doesnt push current into the pi. So no need for a resistor.
You can definitely do this, i did it for a pi connected to my 3d printer. You are wasting the 12V rails though, and some PSUs may not like it if the 12V is unused (YMMV).
When i did this, i got a lot of undervoltage warnings on the pi, but it otherwise worked fine.