This does not really go well together.
You either sacrifice some fund security for convenience and anonymity, or the other way around. The best you're going to get to complete anonymity is a) Bittrex, although they sometimes do freeze accounts without verification when you deposit more than $1k in cryptocoins though i've had more than $1k and never had an issue and b) Shapeshift, which is an instant exchanger, but have higher spreads.
You'll find that most anonymous exchanges will have only altcoins, no USD.
If you want to sell/buy fiats without withdrawal issues, use LBC. Safest and most convenient place around, lots of payment methods supported.