The Q have no more idea of the purpose of existence than any other species. They decided on a course long ago, and have followed it since. For the most part, they don't even think about why any more.
As seen on Voyager, they tend to be very 'hands off', passively observing. They have the technology (perhaps so integrated into their beings that it's more correct to say, 'ability') to travel in time and space seemingly at will, so I don't think it's wrong to assume they do know the future of humanity (as well as most/all other species). It is, however, wrong to assume that when a godly-powered trickster is going to torment someone, he'll tell the truth about what he knows.
They relate to humans...rarely. Only one of them has ever (to our knowledge) sought out humanity. His purpose in doing so varied. Essentially, he wanted to mess with a British-accented space frenchman. He seems to have enjoyed doing so, so he helped Jean-Luc on occasion, giving warnings to him about the Borg.
'Q' enjoyed playing with his toy, so he branched out to other, similar ones. Sisko was different, and much less fun to play with. Janeway was fun to taunt, tease, and heckle. 'Q' seemed to enjoy messing with her much more than Picard (which may have had something to do with viewers liking 'Q' much more than any of the other antagonists, leading to overusing him).