1. Is drinking an extract always a standard action? Reason I'm asking is if I can bypass the full round required for enlarge person. I'm just drinking a vial so there is no full round of casting, right?
Yes, this is one of the alchemist's big advantages.
With the infusion discovery, the entire party can cast your spells with a standard action each.
2. I want to purchase a caster level 20 potion of Stoneskin and use it with the Alchemical Allocation spell so that I never waste it. This gives me 200 minutes duration + 200 minutes from the Extend Potion discovery letting it last slightly over 6,5 hours. Does this really work that way?
Yes, it works this way, but this is totally dependent on the GM being very permissive with the item rules.
A potion of stoneskin must be brewed by a summoner (since this is the only class that gets stoneskin at 3rd level or lower) with the brew potion feat. This peculiar crafting combination automatically makes this an extremely rare item and a GM could flatly state that it's not available for purchase. You've also stipulated an absurdly high caster level, so if your GM allows this he's being extremely permissive.
3. I want to use a Vicious Amulet of Mighty Fists. Will the Stoneskin absorb the 1d6 damage dealt to me? If so I'm looking at some monstrous damage. 2x 3d6+9 claws at +17 are probably already making the other melee characters jelly while I also have bite/gore. The self damage kinda fits the savage theme as well. If I get my hands on a +2 Amulet of Mighty fists I'd make it Vicious and Furious.
The damage from a vicious weapon is untyped energy damage; it bypasses DR.
4. Does Amplify Elixir stack with Extend Potion? My gut says no, but I'm used to seeing a comment that it doesn't stack with X like the Unlife Healer ability of the Undead Lord.
Unlife Healer explicitly calls out that it doesn't stack with other abilities. Neither Amplify Elixir nor Extend Potion have these disclaimers, nor are they worded in such a way as to say this implicitly.
Your gut is probably right as far as the RAI go; this is exactly the kind of thing that normally doesn't stack. However, by RAW, there is no reason they shouldn't. Really up to your GM's call.
5. Do you require to activate the wings discovery to start flying? I remember our GM ruled that the Witches fly hex would take a standard action to start up, but my understanding is that since the wings aren't natural after you fly for x minutes you get tired and need to rest before flying again.
There is a clause that states that "unless otherwise noted, a hex is a standard action", so I think that's where your GM's ruling comes from. There is no such clause for alchemist's discoveries.
I would also add a potion of greater magic fang to use with the alchemal allocation. Buy some boro beads if you don't have room for each of your claws and bite. With extend potion discovery, you only need to use it every other day and can get +5 to attack and damage for cost of potion and extract every other day.