If i walk into a cave or other place with a certain companion, is there are chance they will be like "oh, I've been here before..." and give me a quest or task to do while at that location? Or some other variation???

There are no Radiant Quests that come from the Companions (individual in game companions, not the faction).

I don't remember if there are any followers that give you quests like they did in Fallout New Vegas, however there are quests that you go on where you get ad hoc companions for the quest.