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I'm playing an Alchemist that is taking the Vestigial Arm (Ex) Discovery:

Benefit: The alchemist gains a new arm (left or right) on his torso. The arm is fully under his control and cannot be concealed except with magic or bulky clothing. The arm does not give the alchemist any extra attacks or actions per round, though the arm can wield a weapon and make attacks as part of the alchemist’s attack routine (using two-weapon fighting). The arm can manipulate or hold items as well as the alchemist’s original arms (for example, allowing the alchemist to use one hand to wield a weapon, another hand to hold a potion, and the third hand to throw a bomb). The arm has its own “hand” and “ring” magic item slots (though the alchemist can still only wear two rings and two hand magic items at a time).
Special: An alchemist may take this discovery up to two times.

My question is: would the arms grant a bonus to grapple? The rules for grapple state "Humanoid creatures without two free hands attempting to grapple a foe take a –4 penalty on the combat maneuver roll.". One obvious benefit of the arms means you can leave two hands free to grapple without penalty and use a light weapon in an extra hand. But if you have all four arms free logically I would think you'd be better at grappling. There's no rules I've found dealing with extra limbs, at least not in regards to grapple. If not having two arms free is -4, would it be reasonable to say two extra arms grant a +4?

This is the discovery entry: "The alchemist has learned how to preserve a sample of oozes in a sealed bottle, which he can prepare for use as an extract. When the alchemist activates the extract, he actually throws the bottle at a square within 30 feet, releasing the ooze, which reconstitutes and attacks the closest creature. The ooze is not under the alchemist’s control, but is otherwise treated as a summoned creature. The ooze remains for 1 round per caster level, and decays into powder when the duration expires. If the alchemist has the infusion discovery, another character can use the infused specimen. Creating a bottled ooze requires an extract with a level equal to the ooze’s CR (so a CR 4 gray ooze requires a 4th-level extract)."

What is unclear to me if I need an actual sample of an ooze or if I am creating it from nebulous scratch. The entry starts with "preserve samples", so do I need to kill an ooze to get it? Alive or dead? How many samples does that get me? How long do they each last? How many oozes can I make out of a sample? Or am I over thinking this and the wording is largely fluff, as the entry ends with "creating a bottled ooze requires and extract..."?

I'm playing an alchemist with the vivisectionist archetype, that replaces the bomb class feature with sneak attack identical to the rouge. The sneak attack goes up in level and equals the bomb damage. The Master Chymist prestige class has the Bomb Thrower class feature; "Bomb-Thrower (Ex): The destructive power of bombs appeals to the violent urges of a master chymist. Add the character’s alchemist and master chymist levels together to determine the damage done by her bombs.".

I know normal archetype rules state that replacing a class feature means you do not have it for qualifying purposes, but with this prestige class specific to the alchemist should/could this be made an exception? One could argue that sneak attack would "appeal to the violent urges of a master chymist", and that it levels in damage at the same rate as bombs. Is this something at GM discretion or can I get an official ruling?

No.