Do I need weapons while playing as a Wizard?

All classes effectively use weapons the same way, the only difference is that spell casters don't always visually have them equipped. All damage-dealing spells deal base damage as a multiplier of your weapon damage, and weapon speed affects how quickly you launch your actual spells. You can cast spells more quickly and channeled spells/dots tick more frequently when you are using a faster weapon, and generally hit harder per cast/tick but less frequently when using a slow weapon.

This is pretty easy to see yourself. Try equipping a 1h weapon and casting a spell like magic missile by holding down the cast key, then equip a 2h and cast the same spell. It will be very obvious that the spell fires faster with the 1h weapon equipped.

Sure. The weapons still provide you with stat bonuses. There are even weapons specifically for wizards, e.g. wands and sources. But any weapon will help you deal more damage as many skills deal a percentage of weapon damage (thanks @Arthur). There are endgame wizard builds, which use swords for example.