When cleaning..

[Just Tripp]

does anyone else remove the cylinder and crane from the gun to clean under the cylinder and the gas ring, extractor star, etc? If so, how often?

 

[Bailey Boat]

I usually Keep an eye on the gas ring area and clean when needed but the area under the extractor (what you're calling the "star") is the most prone to fouling. It's exposed every time you open the cylinder and eject fired cases and all the crap still left inside. You can keep a toothbrush handy and brush it clean periodically.
Note: I've seen more revolvers damaged by repeated and un-necessary disassembly than I've ever seen damaged by not cleaning.