I'm trying to build a class around throwing daggers and flasks. To my understand the feat Dual Wielder will not allow you to draw and stow additional flasks due to them not being weapons. I could however, have a flask in hand then throw it and draw a dagger and a whip. Can i only make a thrown attack with a flask if i have Fast Hands, from the Rogue feature? And if this is the case, how do can I draw two flasks in a round?

There's a few things going on here, so let's break it down.

Situation 1: Throwing a flask, then drawing your dagger and whip.

On your turn, you could use your bonus action to use fast hands to throw a flask.

You could then draw both your dagger and whip, so long as you had the Dual Wielder feat. (Otherwise you'd only be able to draw one of them this turn).

You could then move and attack with your weapons.

Situation 2: Throwing two flasks, then drawing weapons

On your turn, you could use your action to throw a flask.

You could then use your bonus action to use fast hands to throw another flask.

You could then draw both your weapons, so long as you had the Dual Wielder feat. Otherwise you'd only be able to draw one of them this turn.

You could still move, but you wouldn't be able to attack.