Quite possible, but I'd say this reputation was damaged recently, when Bitcoin crashed right along with the stock market (granted, its slide started way before), suggesting that it also moves towards the general direction of the economy, rather than against it. It's doing good at the moment, but who knows how long it will last. Gold is still a far better option on that end.
Absolutely. It's not happened often enough, or with any significant consistency, for anyone to be able to say there's a direct and ongoing correlation between the two. At this stage we can only say that there might be some correlation or that yes, there was some correlation that actually turned out to be a coincidence.
I bore in mind when making the OP that, although it is generally accepted that the dollar and gold generally have a very low correlation (one goes up, the other down and vice versa) there are times where this pattern also breaks down.
I think the thing with correlation is that it is simply a statistical measure. It just looks at the price movements of different assets and gives us an idea how closely they follow each other. So we are just applying this statistical measure to bitcoin and seeing what results are being thrown up.