I don't personally buy dresses, but I would say if you like selection, then Villagio is a good place to start - there are lots of dress shops (UK, US, French etc.) Chances are you'd find a store that suits you. The usual suspects include Zara, Mango, Promod, Todds etc.

As for Converse, you can find them in most sport shops. I know Galaxy Sports in City Centre (3rd floor, opposite Quiksilver) carries a small range.

Your best option is to go to Foot Locker in Villagio. (It's near Go Sport - directly opposite The Athlete's foot.) Converse are slightly pricy here in Qatar, but I hope I helped :D