Movement Clarification: A unit may never move off the short sides of a battlefield

Does this rule simply mean that a unit may never move sideways (to the right or left)? If so, can a unit momentum advance into a hex that is to his right or left? If not, does that mean that a unit cannot attack an enemy unit that is in a hex to his left or right?

No, it means a unit can never move off the board from the left or th right side. Sometimes a unit could move off the board from your or your enemy's baseline. A lone leader can move off the board on your side if it's gonna die anyway and you don't want your opponent to get a flag.

Thank you. So just to clarify, the only time a unit cannot move sideways is when it is retreating or evading?

That's correct.