On another note, the outlet diaphragm valve seems to be inoperable. I can't turn it. Is it because it will only operate if you have pressure behind the valve or should I be able to open and close the valve now regardless if it is hooked up to the CO2 or not?
No, you should be able to open and close the valve regardless if it is hooked up to CO2 or not.
In any case though, you will want to remove it to make a cleaner build.
One final question, what exactly is the output range for the Matheson 3813? Is it up to 30 psig or 60 psig?
According to the specification sheet found here, this regulator has a maximum delivery pressure of 30 PSI.