In many episodes with the Q, they seem to place themselves as omnipotent and ultra-intelligent, superior beings - but at the same time, they do such a poor job at portraying themselves as omnipotent, that I'm asking myself how that fits with their "selfproclaimed" superior intelligence.
First of all, a really omnipotent being should be omniscient. Why? Because as an omnipotent creature, you'ld be able to make yourself omniscient.
But they are obviously not omniscient, proven by the many "surprises" they face. This also makes it a bit weird when time travel plots come into play... how can a species that is not bound by time, be surprised by the future? (example: when Q is punished and turned into a human). The result of that is that one can guess: No, the Q are not omnipotent.
Now, why do such incredibly "intelligent" beings do such a poor job at making others believe they are omnipotent, as they seem to imply? If they really are so incredibly intelligent and superior to humans, shouldn't they have a much easier time to manipulate them than just doing some "magic tricks"?
The reasons for this out of universe are obvious (after all, there is still a need to create characters that are entertaining) - but what would be an explanation in-universe? Is it as obvious to other species, that the Q are basically lying? Do the Q, in reality, possess human intelligence just with more knowledge?
And yes, they can access a huge amount of knowledge in a very short time - but that's not intelligence. A computer can do that too.