If I have a Horn of Greed in play and play a Glacial Chasm, do I get to draw before or after choosing which land to sacrifice to the Chasm?
You'll have to perform the sacrifice before you can draw.
If I play the Chasm, but can't sacrifice a land to it, do I still get to draw a card for Horn?
Yes, you still draw. Note that you must sacrifice a land if you have one to sacrifice.
If I have a Chasm in hand and a Fastbond in play (and have already used my 1 "normal" land drop), when I play the Chasm will I take 1 damage from the Fastbond, or will the Chasm prevent that damage?
The damage is prevented.
Finally, can a player "respond" to the playing of a Chasm? I know a land normally can't be responded to, and I assume the same holds true for the Chasm, but I wanted to make sure that the replacement effect didn't open up an opportunity.
No. Playing a land does not use the stack. Neither do replacement effects.