Yes, as the others has stated it is fine to keep the CO2 cylinder in the fridge. Two things of note: First, the regulator will respond more slowly to changes in output pressure when the diaphragm is cold. So if you want to set it at 12 psi, then set it to 8-9 psi and it will 'creep' up the rest of the way to 12 psi.
Second, There is a direct pressure/temperature relationship with compressed CO2. Compressed CO2 is in a liquid/vapor state. As the temperature goes down, so does the pressure. A temp increase will result in an increase in pressure. That is why it is important to keep the tanks away from any heat source, to avoid blowing the relief!
The pressure will stay the same, at a given temperature
, until all of the liquid in the tank has been vaporized. Once only vapor exists, the pressure will drop rapidly. Here is a chart that shows the relationship between pressure and temperature.