Does the order of protonation/deprotonation in the formation of hemiacetals matter? In the mechanism depicted above, deprotonation occurs first and then protonation.
Can this order be changed? If not, why?
The first (productive) step of this mechanism in acidic medium MUST be protonation of the carbonyl.  Alkoxides and hemiacetal-alkoxides have no appreciable concentration in acidic solutions, and the carbonyl is not a strong enough electrophile to be attacked by alcohol without protonation.