Our deepest fear is not that we are inadequate. Our deepest fear is that we are powerful beyond measure.
I thought I read in an earlier post that a player cannot use the advance order to simply move into adjacent hex for movement purposes.
Not sure who said that but it's definitely incorrect as that's exactly what the Advance Order is for: being able to "move" into an adjacent hex but without fear of Op Fire.
Does the player playing advance have to enter into enemy hex and melee
No, he's not forced to but he may.
, exit the board
No, he doesn't have to but he may.
or go over a cliffside?
No, he doesn't have to but he may IF he doesn't possess a Weapon.
He can do no other movements?
Not sure what you're asking here: all units legally activated for an Advance Order may perform that Order.
And for what it's worth, all of the questions above can be easily answered with a quick trip to rule O16 on page 10 of the rulebook.