Does everybody receive the same daily dolmen quest?
I thought this was the case. My first quest was Alik'r and there were swarms of people doing the dolmens. My quest today was Stormhaven and I only met one other person (after zone requesting help) for one of the dolmens. I know that certain zones are better for grinding dolmens, so maybe Alik'r is one of them, hence the large group of people doing them.
Does the amount of players at a dolmen affect its difficulty?
I'm always with other players and there are enemies left, right, and centre of all shapes and sizes, an almost endless stream of them. But after growing impatient in Stormhaven I decided to solo the Gavaudon dolmen. And it was actually really easy (not a brag, I actually do suck at this game lol) - never more than 3 enemies at a time. Same situation with the Menevia dolmen.