Relevant quotes for this question.
At 2nd level, the cavalier can charge across the battlefield at a moment's notice. Whenever the cavalier attempts an initiative check, as long as he rolls an 11 or higher on the die, he can move up to his base speed as an immediate action and he is not considered flat-footed.
If the cavalier takes an action to move during his next turn, he subtracts the number of feet moved during the initiative check from his total movement.
These supple leather sandals grant a burst of speed during times of duress.
When the wearer acts during a surprise round, he can take a standard and a move action during the surprise round. If the wearer already has the ability to take a standard and a move action during the surprise round, he instead gains a +10 circumstance bonus to speed when acting during a surprise round.
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I'd say 1) happens. Initiative checks are made before the surprise round.
At the start of a battle, each combatant makes an initiative check. An initiative check is a Dexterity check. Each character applies his or her Dexterity modifier to the roll, as well as other modifiers from feats, spells, and other effects. Characters act in order, counting down from the highest result to the lowest. In every round that follows, the characters act in the same order (unless a character takes an action that results in his or her initiative changing; see Special Initiative Actions).
If two or more combatants have the same initiative check result, the combatants who are tied act in order of total initiative modifier (highest first). If there is still a tie, the tied characters should roll to determine which one of them goes before the other.
If Initiative checks were part of the surprise round, then there would be no way to decide the order characters acted in the first round.
Initiative is rolled
those who can act in surprise rnd do so in correct initiative order
then combat proceeds as normal.
personally i hose rule it that EVERY encounter has a surprise rnd, even if no one can act.
I see it going something like this ...
GM: Everyone roll a perception check
PCs: Announce rolls, none good enough to detect enemy, thus, there will be a surprise round.
GM: Roll initiative
PCs: All roll, Cavalier gets > 11.
Up until the Cav's turn in the surprise round comes up, he can use his order ability to do what it does (move as an immediate action).
When the Cav's turn comes up, THEN the sandals function. He gets a standard and a move action (NOT a full action). If he had used his order ability, he is considered to have already spent his Swift action and his move action is reduced by whatever movement he already used.
So my answer is mostly 1.
Initiative is rolled
those who can act in surprise rnd do so in correct initiative order
then combat proceeds as normal.personally i hose rule it that EVERY encounter has a surprise rnd, even if no one can act.
That seems dangerously favourable to Divination school wizards, or Oracles with the Temporal Celerity revelation, or others who always act in the surprise round. Could really mess with action economy.