What happens if Baron Rivendare and Dreadsteed die at the same time?

The simple answer to the question is that you get 1 Dreadstead back on the field.

Using your example, I'll use Twisting Nether as my board clear (Though any other board clear will have the same result).

Only the single deathrattle from the dreadstead will occur. Your Baron is not "alive" to trigger the deathrattle a second time.

The first set of pictures show Dreadstead played first.

This set shows Baron Rivendare played first.

As you can see in both cases, only the single deathrattle from each Dreadstead takes effect because there is no Baron alive to trigger the second one.

For some minions, their effect does not trigger if they die.

For example, you have a Cult Master (draw a card when a friendly minion dies), and 4 different 1-health minions.

Your opponent (Paladin) plays consecration, killing all your minions.
Result: You draw no cards

If your opponent is a mage, and plays Arcane Explosion instead, all your minions but the Cult Master will die.
Result: You draw 4 cards (one for each minion)

Although I haven't had extensive experience with Baron Rivendare specifically, I assume this to apply similarly. Rivendare's power does not work when Rivendare dies.

EDIT Addition to the below comment thread

• Deathrattle triggers when the minions die. Order of deathrattles = order of said minions being played. Deathrattles that summon minions do so after the current action has completed.
• Triggered effects (lightning bolt) only trigger when the appropriate condition is met and the minion with this power does not die during the same action.
• Active effects (no lightning bolt, e.g. Stormwind Champion or Dire Wolf Alpha) are always active and only deactivate after the minion with said power dies.

I'm not sure about Inspire, as I haven't seen too many inspire plays yet.
My expectation is that the Inspire action only triggers after the hero power is used (and its effect has applied). So the order of operation would be:

1. Play inspire minion. ("Inspire: This minion gets +1 attack")
2. Use hero power (let's say Mage)
3. 1 damage done to target (from mage hero power)
4. Minion gets +1 attack.

This would be analogous with how e.g. Lightwarden works. Their attack gain only occurs after the rest of the action has been resolved.