If the draw part of the effect is skipped due to a replacement effect, the discard part of the effect is not skipped.

If multiple effects modify your hand size, apply them in timestamp order. For example, if you put Null Profusion (an enchantment that says your maximum hand size is two) onto the battlefield and then put Anvil of Bogardan onto the battlefield, you'll have no maximum hand size. However, if those permanents entered the battlefield in the opposite order, your maximum hand size would be two.

The triggered ability is put onto the stack after you have already drawn your card for the turn.

If there are two Anvils on the battlefield, they trigger and resolve separately. This means that you first draw one card and discard one card, and then do it again. You do not draw two cards and then discard two cards.