"I want the biggest cookie."
Translation:Jag vill ha den största kakan.
Swedish adjectives in definite superlative generally have an additional -e ending. This particular case is an irregular exception where it's -a instead. But it has nothing to do with the plural -a of e.g. the base form "stor". (Note though that störste would actually also be correct when describing males - this is a remnant from when Swedish had a more thorough gender-based adjective system.)
Well, it's a completely different construction - corresponds to "I want the cookie which is the biggest". And you have a few typos, it should be Jag vill ha den kakan som är störst, although you could also use indefinite kaka and/or definite den största - Swedish is a bit flexible here.