Incorrect Ring Flip? Is the flip of the cis-isomer wrong? Specifically the image on the bottom right. 



What I see is that on the cis-isomer in the first chair, we have an axial upward methyl group and an equatorial upward methyl group. When the ring flips we now have an equatorial upward methyl group (good) and a downward equatorial methyl group (bad). Shouldn't the second one be axial upward? From what I understand, ring flips shouldn't change the orientations of groups; just their axial/equatorial configurations. 

Also, in general, would these two rings be equivalent?
Yes, the cis isomer on the right should have the methyl group on C2 in the axial (up) position.  The two rings you've drawn at the bottom of your post are equivalent, they can be interconverted by a 180 degree rotation about an axis that bisects the C1-C2 bond and the C4-C5 bond.