Willdabeast314

Have a look at using 9X raspberry pi and one or two full PCs for gaming. The current model isn't a bad little Linux based web browser and media player. raspberry pi + case + power supply + SD card can be done for under $100 each. You might find through your church you can get some of these items donated. Head over to the Raspberry pi forums, they're very friendly over there and someone will be able to tell you how they set up their school/church:
https://www.raspberrypi.org
 
 

As its a church lab not running a mission critical 24/7 corporate database i don't think you need a Xeon. I see you haven't gone for ECC RAM, another reason you may not need xeon.
16 Xeon cores /9 = 1.7 cores per desk not counting server host
16GB of GPU = 1.7GB GPU RAM per desk
3.5Gb ram per desk. 
You can find cheaper keyboards.
Your base price of monitors, keyboards, mice, headphones is $2208.24
that leaves you $532 per desk to spend on your server + thin clients or 9X full computers.
I don't like the lag for office applications in thin client solutions and I wouldn't use it for gaming. IMO I don't know if you need a central server & storage

What i have been experiencing with during my summer employment is using a raspberry pi to remote into a VM, ive done this through citrix and RDP. Ive only done this using Nutanix servers, but your build looks good enough. My question is how are you going to provide sound and video to all the monitors? Are you going to do one of those awful setups (IMO) that linus did in 7 gamers 1CPU?
 
I could go on for hours talking about how to do what you want to do properly, but sadly im not getting payed thousands of dollars to so theres my quick rundown.
 
GL!