I don't use a perception skill. Basically, perception is a component of an active skill: if a character wants to detect someone sneaking by, this is a contested Stealth skill check. If they want to know if someone is lying to them, it is a contested Befuddle check, and so on.
I do not use rolls for passive detection (i.e. for 'clue detection'): I use the value of the corresponding background skill to determine if they notice something noteworthy or unusual or not (e.g. Architecture to notice that the interior of the building does not match the exterior, suggesting the existence of hidden rooms or passages). The level of the skill indicates how much they notice (non-active skills are on a different scale and are not used in rolls).
Given the above, the reroll question is typically N/A: if the perception is active or reactive and has been missed, the situation has evolved and does not warrant a perception attempt anymore. If the perception is passive, there is no roll, so no reroll either
All rolls (perception or otherwise, including combat) are resolved the same way: roll die (I use a d20 instead of a d100, but it basically works the same way, with 1 being LOE and 20 being HOE), add modifiers, compare with threshold : 21+ indicates a success, 20- indicates a failure. margin of success/failure indicate quality (21-25 is 'yes, but', 26-30 is 'yes', 31+ is 'yes, and' / 16-20 is 'no, but', 11-15 is 'no', 10- is 'no, and').
I don't know what 'snap perception' means: either perception is active (player action), reactive (automatically triggered by an external event or intent, such as an assassin trying to sneak close to slip a blade in), or passive (the character looks around and notices something in the static environment or situation).