I play timetwister, and in response I fork it. First, the fork resolves, and I shuffle in the timetwister along with the rest of the cards and draw seven. Then, the timetwister resolves, and I do it all again, shuffling in the fork. But according to the text on timetwister, the card should be placed in the graveyeard: Do I then fish it out of the deck and place it there, or do I finish with an empty graveyeard?
Not exactly either. The reminder text "(Then put Timetwister into its owner's graveyard.)" (C.R. 108.1) is intended to remind you that Timetwister is not in the graveyard when players shuffle their hand and graveyard into their libraries (rather, Timetwister is on the stack at that time); like nearly any other resolving spell, Timetwister goes to the graveyard when it finishes resolving (C.R. 608.2m).
Here is how this situation can play out in general:
Assume you have Fork and Timetwister in your hand and you control Underworld Dreams. Assume also that it's your main phase and nothing is on the stack.
You cast Timetwister; then, retaining priority, you cast Fork targeting Timetwister (C.R. 117.1a, 601.2i).
All players pass, then Fork resolves. You copy Timetwister, then Fork goes to the graveyard (C.R. 608.2m). (Note that unlike what some printed versions of the card may suggest, Fork doesn't become a copy of the spell it targets [C.R. 108.1].)
All players pass, then the copy of Timetwister resolves. Each player shuffles their hand and graveyard into their library (so that the Fork in your graveyard is shuffled into your library) and draws seven cards (and Underworld Dreams's ability triggers once for each card drawn this way by an opponent of yours). Then the copy of Timetwister goes to your graveyard (C.R. 608.2m) and ceases to exist because it's a copy of a spell (C.R. 117.3d, 117.5, 704.5e).
The Underworld Dreams abilities go on the stack. For each ability, all players pass, the ability resolves, and the opponent it refers to is dealt 1 damage by Underworld Dreams.
All players pass, then the original Timetwister resolves. Even though each player's graveyard is now empty, each player still shuffles their hand and graveyard into their library (C.R. 701.20d) and draws seven cards (and Underworld Dreams's ability triggers once for each card drawn this way by an opponent of yours). (See also this thread.) Then the original Timetwister goes to your graveyard (C.R. 608.2m) and stays there since it's a card.
The Underworld Dreams abilities go on the stack. For each ability, all players pass, the ability resolves, and the opponent it refers to is dealt 1 damage by Underworld Dreams.
I have underworld dreams on the table and timetwister and fork on my hand.
I play timetwister, and in response I fork it. First, the fork resolves, and I shuffle in the timetwister along with the rest of the cards and draw seven. Then, the timetwister resolves, and I do it all again, shuffling in the fork. But according to the text on timetwister, the card should be placed in the graveyeard: Do I then fish it out of the deck and place it there, or do I finish with an empty graveyeard?
Here is how this situation can play out in general: