No, it happens anytime a formation enters an hex where an enemy is present BUT only if one of the formations wants to actually engage the enemy. If neither wants to engage, no check is made. In that case, and if MP are left, the movement goes on with a +1 penalty to exit the hex.
Is there an errata on that? Your statement does not match the rules as written, as the rules state that "Any time a Formation attempts to move through or out of an [sic] a hex containing an enemy formation, an Engagement Control Roll may occur to determine if combat unfolds or whether both Formations continue to move as normal." I get that that makes sense, but the rules seem focused on the act of leaving a hex, "through" or "out", being the trigger, not entering.
Didn't understand what you mean here.
If there are 4 formations on the map, 2 for each player, the initiative order will be A-B-A-B where A is the one that lost initiative. If on their first move, play A moves a formation onto one of B's formations and an engagement occurs, "both Formations are
engaged—their movement ends immediately and they do not act again until the Combat Phase". So now player B has their turn and only one formation left with a movement phase, are they required to move that formation or can they "sink" their move by attributing it to their formation that no longer has movement?
No. You cannot move outside your movement turn. And Overrun is NOT "something you do during the engagement control check". Overrun is a special attack you do when you want to enter and exit an hex occupied by enemy WITHOUT having to stop there and make an angagement roll. If the overrun is successfull your movement goes on, if the overrun fails your are stuck there AND engaged.
That does not seem to mesh with the rules, pg 168 Overrun section "Make an Engagement Control Roll as normal, with the following modifier: compare the Size of each Formation and apply any difference as a modifier to the Engagement Control Roll." Likewise, the example on that page makes it clear that the Overrun action is part of the Engagement Control Roll, not a second roll or subsequent action of some kind. The issue is less with Overrun, it makes some sense that you can't counter charge, but Evade also requires that you move out of sequence. If Overrun and Evade are not available to the responding player during an Engagement Control Roll, that more or less means the attacker controls engagements as the responding player has no response to avoid an engagement (overrun or evade) like the attacker does.