No need to cook mushrooms or onions. Just mix it all in a bowl and refrigerate for 12+ hours before serving.
It's all about the vinegar.
Any onion tastes the same in this recipe, but red onions just look better.
With this vinegar/oil proportion, it'll taste exactly the same no matter what mushrooms/vegetables/whatever you put there. To make the mushroom/whatever taste more distinct, substitute some of the vinegar with water.
You can add minced garlic. You need a lot of it for it to make difference (7+ cloves).