So you need to move past the fear of breaking it since it's now broken in some way. Unless it was broken you usually don't need to pull off the heatsinks. I do when I think replacing the compound could clear up a failure but otherwise why do this removal?
If you must remove this your mechanical skills will be tested. How many times have you drilled out a bad screw, tapped new threads or cut a slot in a screw head so you can get a screw out?
There is no exact method you should use here. The choice is yours as a mechanic.
I have a HP dv6-6042eo. When dissembling it, I run into a problem - two heat sink screws with springs on them are stuck. They are spinning in place and not moving or somehow stuck. And the golden gear parts on the other side of the screw hole are moving to and that is why I think the screws are sppining in place. I don't know what to do and don't want to break the motherboard.
Any suggestions?
I'll leave a link for reference what I'm talking about (the 8th and 5th screwa are the problem for me)
Reference (two upper screws on the right):
http://www.myfixguide.com/manual/wp-content/uploads/2014/11/HP-Pavilion-DV6-disassembly-16.jpg

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic