Say my warriors charge and engage 3 out of 6 models from an enemy archer squad. Obviously during this round, the engaged models must now fight back, hold their action, or become exhausted.
Is this also the case for the models in the archer squad that weren't engaged?
Say the next round, my opponent has the initiative and he activates this archer squad first. Can the three unengaged models still shoot while the three engaged models fight in melee? Or is engagement really on a squad level?
Engagement is the term used to describe individual models in base-to-base contact with enemy models. Any squads with Engaged models are considered to be in Melee Combat.
As the archers' squad is in Melee Combat, they cannot fire ranged weapons. They would either have to remain as they are, or jump into the melee themselves. It's worth remembering that enemy squad can move up to their base Movement in melee combat if they would be able to tag additional models.
As an aside, ranged weapons with Threat Range and anything with the Pistol ability can be fired in melee, targeting the enemy's MTN and using your Ranged to attack instead of Martial.