It is possible keep items created with spheres of might, even when I change the talents as armiger

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.

Long story short, you cannot. All ways to "modify" it will also involve permanently removing all magic from the item.

Most magic dispelling effects are temporary. A Disjunction spell will destroy the item completely. And breaking it until all hit points are lost will remove all magic, but the item can be fixed with spells and remain a simple masterwork item, meaning that it's much easier to simply buy another masterwork item.

As for mundane items, you will have to ask your GM, as there are no rules for this on the Craft skill. And most modifications will be a case by case analysis.

Like, attaching a chain to a kama will make a kasurigama, attaching another axe head to a two-handed axe will make a double-axe. If you make a pointy tip with metal, bone or otherwise and tie it on top of a quarterstaff, you could call it a spear. And so on.

But a lot of items will simply not work that way, and each GM/table will have different opinions on each item individually. Like, what is the difference between a composite bow +1 and a composite bow +4? What makes one stronger than the other?

I would say "everything", the craftmanship is completely different, you need a different wood, a different technique, etc. But some other guy might understand more about crafting of bows and say "no, all you have to do is work on it for a little longer and it will be studier". So this will suffer from table variation.

This kind of modification could easily fall under the Repair Item rules, as the base item isn't changed and is simply being modified, and as such:

• Pay 1/5 of the cost of that new item.
• Make a craft check against the same DC to create the new item.
• If you fail by 5 or more, the cost paid in materials is wasted and you must try again.

You might consider the difference between the two items, if they are too different (new item is more expensive than the base item), you might consider that you must make a new item altogether, and use the base item cost as material cost instead.