Down to one Doom token. I fail my Combat roll. Can my friend use Carcosan Page to take my Unique Item and use it in his combat roll before I get devourred?

I don't actually know that one. I would assume that, like a regular combat, the round has to be resolved before the Page can be used. So that means that the only time you would be able to use it is Upkeep, but that's when you can regularly trade items anyway, which would render the CP effectively useless.

I would understand that. It's just that the Item being used to trade items says that you can use it during any phase to freely trade items between players. The game is hanging on the balance at this point. We rolled in case the ruling was that we were aloud to trade items using the Unique item. If we can then we win. If we can't then we lose lol.

No you cannot based on how you phrased this.

One of the rules of thumb in Arkham is that once an "action" starts, it goes to its completion. There is very little that actually interrupts an action. So as your one investigator is dying, you couldn't interrupt with CP and grab the item. You would have to wait for that round of combat to resolve. Once it dis, the investigator is devoured and his items discarded before CP has a chance to kick in.