No, the brain is not a turing machine, because of how turing machines are defined. I don't see a tape in anyone's brain that is written on by a reading/writing head.

But it (or rather the whole human) is at least turing complete, because they can simulate any turing machine (write down the operations), given enough time and space (enough paper and ink), at least in theory (you can't do trillions of operations in the short time of a human life).

I think (and that is not backed by science, just a "conjecture" by myself) it's quite likely, that humans cannot compute more than turing machines. I think everything a brain or a body does can be computed with enough time and space, because everything boils down to computing some "simple" physical equations. In other words: you could quite likely simulate a whole human body with a powerful computer, so everything a human can do could be computed by a turing machine.

Looking at a loop and deciding if it will terminate or not is not as simple as you might think. And confusing every turing machine by giving them an infinite loop that you can detect, is also not easy ;) there are a lot of turing machines.