If someone pays the cost, then the card was never drawn or discarded. It just gets put into the graveyard directly from the library.

This is continuous replacement effect that modifies a draw.

If a spell or ability causes more than one card to be drawn, draw the cards one at a time and deal with this replacement before drawing the next card.

When using Dredge with this, you have two choices. You can immediately replace the draw with Dredge, or you can apply Zur’s Weirding’s replacement. If the latter, you first resolve Zur’s Weirding’s ability, then if you’re going to draw the card, decide whether or not to replace that draw with Dredge.