The whole shroud comes off. Only a couple companies officially allow you to change the coolers, but require the stock cooler when RMAing.
As far as advisable, I've replaced thermal paste which required removing the cooler and it helped the temperatures. I wouldn't do it on the brands which will not allow you to do it, it'd just be an excuse to deny an RMA.
Performance will not be increased by changing a cooler in itself. You may gain by lowering temperatures, therefore allowing you to overclock further (provided you don't hit the actual GPU max before the temps rise to far), but with e.g. the NV gtx 6xx cards you can't touch the voltage so the value is less. Generally a custom cooler is cooler, thus allowing higher voltages, overclocks etc. and still having a lot quieter experience then the normal reference coolers. I wouldn't change coolers on a gtx 6xx card personally because they seem to hit their limits with the stock coolers being you can't touch the voltage. I would consider it for a HD 7xxx card if you don't max it out and it starts getting hot.
The cost of the aftermarket coolers is generally pretty high so you may as well buy a custom cooled card. (HAWK/Lightning, DCII, etc.)
The manufacturers. If you find more 'official' info that is missing/incorrect let me know.
https://docs.google.com/spreadsheet/ccc?key=0ArCtGVODPUTtdEdEUjRiSFdyckZ1Q1dGNUI3bkd5R1E#gid=0