The compressors do come on each time you start it. If the sensor does not sense a load then it only comes on for about 5 seconds to check the system. If it reads a load then it will come on long enough to fill the shocks and get the load sensor back to where it should be. If there is a leak it will run for a while and turn off. I do not remember how long it runs before it shuts off. It shuts off to protect the compressor.
There are several replacements available. I may have the terms "active" and "passive" backwards as I have seen them used both ways but, there are active replacements that work just like the originals and passive ones that use a resistor to fool the computer. Both passive and active are air shocks that will work with the leveling system. Passive shocks do not work with the stability control which is why the computer needs to be fooled.
Arnott is popular. They have both versions. The active replacements are rebuilt originals with new air bags. I am not sure if the do anything with the original magnetic dampening fluid in the shocks. I believe they also have passive conversion kits. Up here in the north I have heard folks that are not too happy with the rebuilts. They say they won't hold up in the cold weather. Not sure myself.
Suncore also has replacement shocks. I have found that some of the major shock companies like Monroe do make active replacements but they were near $400 each last time I checked. Have not heard of anyone using them. There are even passive sets on ebay that sell for less than $150 a set but I have heard from a couple of people that they do not last long.
Lastly, there seem to be a lot of folks that dump the air shock system and go with better standard shocks. There is a lot of info on this site about conversions.
Most of the info I have found is because my truck still has the original system which just started leaking this winter at about 180,000 miles. However the ride has seemed harsh since I got it with 165,000 so I started looking for replacements. I was told they get hard as the magnetic fluid breaks down which is why I am hesitant to get rebuilts unless the fluid is also replaced. I think most of the info I have is fairly accurate. If not, hopefully someone will correct me.