If already in the engaged state, can a unit declare CC attack with the first short skill, and declare Move with the second short skill, allowing them to move if they win the F2F, or resolving to idle if they don’t?
Because you wouldn't make rolls until after validity of declared skills is checked you would still be in the Engaged state so the Move would automatically revert to Idle before the F2F is made.
Troopers in this State can only declare CC Attack, Dodge, Idle, Reset, and Skills that specify that they can be used in Close Combat or in Engaged State. (See FAQs & Errata. Berserk can be declared while in Engaged State.)
Only if the attacker is within ZoC, and they will take a -3 penalty for not having LoF to the attacker. 6th Sense would allow the target of an attack to Dodge at no penalty against an attack from any range.