The first ability triggers when Golgari Brownscale returns to your hand from your graveyard for any reason, not just when you use its dredge ability.

You can't attempt to use a dredge ability if you don't have enough cards in your library.

Dredge can replace any card draw, not only the one during your draw step.

If an effect puts a card into your hand without specifically using the word "draw," you're not drawing a card. Dredge can't replace this event.

If you're drawing multiple cards, each draw is performed one at a time. For example, if you're instructed to draw two cards and you replace the first draw with a dredge ability, another card with a dredge ability (including one that was milled by the first dredge ability) may be used to replace the second draw.

One card draw can't be replaced by multiple dredge abilities.

Once you've announced that you're applying a card's dredge ability to replace a draw, players can't take any actions until you've put that card into your hand and milled cards.