Does the Surprise Round count as the first round of combat for the purposes of Advance Warning?
Advance Warning
You can shout warnings to nearby allies, preventing them from being caught off guard.Prerequisites: Cha 15.
Benefit: All allies within 15 feet of you are only flat-footed during the first round of combat until your first turn. This does not prevent them from being flat-footed due to other situations, like being flanked. In addition, all allies within 15 feet of you gain a +1 dodge bonus to their Armor Class against all ranged attacks. These benefits only apply to those allies who can hear and understand you.
The Surprise Round: If some but not all of the combatants are aware of their opponents, a surprise round happens before regular rounds begin. In initiative order (highest to lowest), combatants who started the battle aware of their opponents each take a standard or move action during the surprise round. You can also take free actions during the surprise round. If no one or everyone is surprised, no surprise round occurs.
Unaware Combatants: Combatants who are unaware at the start of battle don't get to act in the surprise round. Unaware combatants are flat-footed because they have not acted yet, so they lose any Dexterity bonus to AC.
Emphasis added.
Argument against: CRB says the surprise round is before regular rounds. The terminology in Advance Warning can be interpreted as meaning it applies to regular combat rounds, as it doesn't mention surprise rounds.
Argument for: Shouting a warning is talking, which is a free action, which CRB says is allowed during a surprise round. The language 'before regular rounds' implies that the surprise round is a combat round, just not a regular one. Warning an ally that they're about to be attacked seems like exactly the sort of thing this feat is designed to do, implying RAI.
To clarify, the 'argument for' is only so far as allowing them to avoid being flatfooted. In order to actually act, you'd need the Lookout (Teamwork) feat. This is just about AC bonuses and sneak attacks, essentially.