Do its bonus stack with Aid Another's not-circumstancial bonus ?
Yes. They are bonuses of different types, and therefore stack. Untyped bonuses always stack with all sorts of typed bonuses, and also with most other sorts of untyped bonuses (the exception being that a bonus will not usually stack with itself if you would be entitled to it more than once, even if the bonus is untyped; this is known as the Same Source limitation). They could have made it an exception if they wanted by explicitly saying it does not stack with Aid Another, but that would be pointless. Thus someone using Aid Another while having the feat effectively doubles the bonus that they offer.
What does mean "your assistance doubles the gp value of items that can be crafted each day” ? Do you craft twice faster (thanks to a very twister way of reading "double the gp value you can craft each day") ?
That’s precisely what it means. Actually, you craft a little more than twice faster because you are hitting a higher DC thanks to the feat’s +2 circumstance bonus, but that particular line doubles your speed compared to if that line was not there. Your progress on any crafted item is measured in silver; when the silver value of your work crafting the item is equal to the value of the item in silver, you are done. If your crafting checks produce twice as much silvers’ worth of progress, then you will meet the item’s value in half the time.
Why is Cooperative Crafting a feat?
A good question; to my mind it’s not very good design. Aid Another was already sufficient, I think, to model someone helping you out. If anything, I would have made this sort of bonus contingent on hitting higher DCs with your Aid Another roll (since, as you say, DC 10 is remarkably easy to hit), and for having both people have the relevant feat. This seems like an unnecessary feat tax to me, even if the effect is reasonably good.
On the other hand, I’m not too familiar with Pathfinder’s options in this regard, but if there are enough effects out there that stack and reduce crafting times or crafting costs, that can lead to very overpowered characters if they can get them all (see 3.5’s Artificer). Which makes me leery of an effect that doubles crafting speed, particularly when you don’t need to take the feat yourself; a cohort could take it, you could hire an NPC who has it, or whatever. Alone, this feat is fairly week. In combination with other, similar feats, it might be overpowered. Which again would just lead me to wanting to have a lesser effect, achieved just by hitting higher DCs than 10 on Aid Another.
Yes. Lab description (find it here) says:
This lab is used for making alchemical items, and provides a +2 circumstance bonus on Craft (alchemy) checks. It has no bearing on the costs related to the Craft (alchemy) skill. Without this lab, a character with the Craft (alchemy) skill is assumed to have enough tools to use the skill but not enough to get the +2 bonus that the lab provides.
So yes, even without common sense laboratory = equipment, description of what lab is ensures that it is in fact sufficient, and that it not only allows you to use alchemy, but also gives you bonus.
Equipment parts and supplies that gets used up / damaged regularly, like oil for heaters, charcoal in filters etc is included in material cost of alchemical creations already. Treat them as raw materials, because complicated, hard to get by things are made to last. And for other things like broken vials and flasks - we don't care for nicks on swords, we ignore minor damage on armor, why would we care for minor wear on lab?
Best Answer
The alchemist's lab specifically overrides the more general masterwork tool and the masterwork artisan's tools
Below are relevant passages from the Player's Handbook. They don't make a lot of sense when considered individually, which is why I included the above summary.
The Player's Handbook on Favorable and Unfavorable Conditions says
Emphasis mine. The Player's Handbook later describes the masterwork tool as follows:
Emphasis mine. Note that the list of examples isn't exhaustive and that Craft is a skill. The Player's Handbook also describes artisan's tools:
And masterwork artisan's tools:
Emphasis mine. Finally, the Player's Handbook describes the alchemist's lab, in part, as follows:
Emphasis mine. When a skill lacks a specific perfect tool, you buy a generic masterwork tool or, for a Craft skill, generic masterwork artisan's tools for that Craft skill. The skill Craft (alchemy) has a perfect tool available: the alchemist's lab. This makes both a masterwork tool and masterwork artisan's tools unavailable for the skill Craft (alchemy) in the same way you can't buy a masterwork tool for the skill Heal instead of a healer's kit. To get the +2 circumstance bonus for using the perfect tool on Craft (alchemy) skill checks you use an alchemist's lab.