Conformations of cyclohexane and flipping
Essentially, it is stating that a ring flip retains the absolute stereochemical configuration. Ring flipping is a physical phenomenon of 6-membered rings, not a chemical reaction that interconverts stereoisomers. No bonds are broken, though many do get strained in the process.

I'm not sure what you mean by your question about the importance of being in the axial/equatorial positions. Both the methyl and chloro groups are vying to reduce 1,3-diaxial groups by placing themselves in the equitorial position, but doing so for one places the other in the axial position. I think a natural question that arises "is which one wins?"