int chk = DHT.read22(DHT22_PIN);
The read22() call tries to read data from the sensor and fills the variables DHT.temperature and DHT.humidity. The return value tells if something went wrong during that "exercise". Some people work with the "good weather" scenario and leave out all the checks, that's a design choice.
When I'm asking the sensor for the current humidity and temperature, are those two different requests?
NO, if you do a read22() both temperature and humidity are read. They can be obtained by reading the members DHT.temperature and DHT.humidity
Or does the
int chk = DHT.read22(DHT22_PIN);
request call all the sensor information, and load it into the Arduino memory, waiting for the DHT.temperature and DHT.humidity calls?
Yes, if you dive into the code of the library (don't be afraid 
, you see that temp and humidity are both filled in the read22() call.
MY errorrate with this sensor was definitely lower, no figures at hand.