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as a GM (often of PFS) I wouldn't allow this - if you can't act in the Surprise round that indicates you aren't yet aware of the combat - I wouldn't let you do any actions as part of that combat at all - including "free" actions such as resolving a knowledge check and communicating the results to the other players. There are a few classes that get special rules here (Diviner Wizards - i.e. with the Divination Arcane School may have an ability to always act in a surprise round - i.e. they are magically aware of combat before it happens)
similarly if you are flat-footed that means you haven't yet reacted to the combat - here there are more classes that have exceptions to it (any class with uncanny dodge for example can't be caught flat-footed) or someone with Combat Reflexes. As a GM (and as a player) think it is reasonable to treat this case of flat-footed (not yet having reacted to combat) similarly to a surprise round - you aren't yet ready for the combat. This also has the effect of giving characters with uncanny dodge or combat reflexes another small bonus (ability to speak and react first) which seems both matching the flavor of those classes and a simple yet fair way to handle things for all involved.
From a practical matter it is easiest to resolve knowledge checks in combat on a given player's turn - for one events in the battle may change what checks a player can make and what they choose to do with that knowledge (i.e. allies might kill one enemy, an enemy may flee or some abilities/features of the monster may become clearly obvious to all as a result of actions taken). Aa well though the rules don't actually allow this (since a knowledge check represents that you know) most GM's let multiple people who succeed on a knowledge check against a given enemy each ask their own questions/ gain different knowledge about the enemy (how this is resolved is actually left to the GM - some I know tell Players what those players know while other GM's let players choose what questions they have about the enemies.
Most Free Actions actually can only be taken on your turn (speaking is an exception which is specifically called out as something you can do on any turn) But remember as well that you can only say a few sentences - I've seen many many tables where people have long form conversations while in the middle of battle, forgetting that the speaking part of a free action represents literally a second or so to shoot something to your allies - it isn't time enough to dictate detailed instructions.