They're not Controlled - It's CONTINUOUS Descent Approaches
1) Where I work - yes if achievable. All operators are monitored for CDA compliance and follow up action is taken in terms of communications with the company if CDA compliance falls below a certain level.
2) Neither - see above
3) CDA's tend not to be refused as such. But some airlines are better at doing them than others.
For us it's a distict advantage as you no longer require a "stepped descent" to keep the aircraft inside CAS. You give the aircraft the track miles to run and then start the descent using the "3nm per 1000ft" rule. The resulting profile if correctly flown will keep the aircraft inside CAS with no further input from ATC.
Sad to say some people still haven't grasped the concept, and still push the nose down to 2500fpm when half that is required. There seems to be a belief from some more "experienced" pilots that DME distance overrules the miles that they're given by ATC - this is one part of the culture that needs working on. The introduction of FMS flown P-RNAV approaches will be a great help in this respect.