If a player has a Unit card - Doomsayer of Morr (Action: When an opponent's unit is destroyed, draw a card) in play and another player plays a card that destroys all units, does the now dead Unit - Doomsayer card get to draw on the opponent's destroyed units from this play if they are destroyed as well.

The main question is when does the action of the card take place, if it is being destroyed at the same time can it still play the action?

If Your opponent plays a card (Troll Vomit, for example), and destroys Your Doomsayer, i don't think You get to activate its power (a triggered action).
Let's say the opponent plays the card abovementioned: if i use the Doomsayer action in response, when the action chain starts resolving we have the following situation, based on the LIFO structure:

1) Doomsayer's action: however, the opponent's units are all still there, there's no triggering condition so no card is drawn
2) Troll Vomit action: all units, including the Doomsayer, are discarded.

If i don't use the sayer's action and wait for the opponent's action chain to resolve, the unit is simply destroyed and there's no power to activate to start a new action chain.

No, the Doomsayer has to be in play for its trigger to activate. When multiple units are destroyed by the same effect, they all leave play simultaneously, so the Doomsayer is already gone when the death of another unit could trigger him.

Statement above is correct, but remember that IF other units than Doomslayer leaves play due to same effect (like Judgement of Verena) then you draw card for each unit that left play