Answering as a recipient:
1) U-Haul has a great variety of quality boxes. This sounds obvious, but make sure there's AT LEAST one inch of clearance all the way around, which should be filled with packing material. If the box is bulging, you've done a bad job.
2) I've had two sets shipped to me by UPS with no issues. I can't speak to how they compare on price, but they seem to be the most willing to follow instructions regarding where to leave stuff. For my most recent set, I left a tarp out because it looked like rain and the driver did a great job in using it (bonus: he found a way to keep the boxes off the ground).
3) I would rather receive un-nested drums, but I understand that this isn't always possible. For packing material, I prefer wadded-up newspaper or kraft paper. While packing peanuts also work well, they leave crumbs that can take time to remove, especially if they've somehow acquired a charge during the ship.
4) This is not the time to economize. Use good boxes, plenty of quality tape, and permanent markers. You get what you pay for.