1) Yes, a unitary transformation like the Hadamard gate maps the state |0> to |0>+|1>, while mapping |1> to |0>-|1>. In either case, if you then just measured immediately in the {|0>,|1>} basis without doing anything else, you'd see |0> or |1> with equal probabilities, so it would have the effect of a coin flip. But of course, in other cases---e.g., if you measured in a different basis, or if you applied the Hadamard to a state that wasn't just a |0> or |1> basis state---you could see that Hadamard is not just a coin-flipping transformation, because it's able to produce interference.

2) When we talk about the different "paths" that contribute to a given amplitude, it's just a fancy way of saying that we can organize the matrix multiplications in such a way that the amplitude we want is a giant sum. So for example, suppose we apply Hadamard twice in sequence to the initial state |0>. The first Hadamard maps |0> to (|0>+|1>)/sqrt(2). The second Hadamard maps |0> to (|0>+|1>)/sqrt(2) and |1> to (|0>-|1>)/sqrt(2). So by linearity, it maps (|0>+|1>)/sqrt(2) to

((|0>+|1>)/sqrt(2) + (|0>-|1>)/sqrt(2))/sqrt(2) = (1/2+1/2)|0> + (1/2-1/2)|1>.

So in this case, we could say that there are "two paths leading to |0>," both of which contribute 1/2 to its final amplitude (so that the amplitude is 1). There are also "two paths leading to |1>," but one contributes 1/2 to its amplitude and the other contributes -1/2, so the two contributions interfere destructively and the final amplitude of |1> is 0.

This is sometimes called the "Feynman" or "sum-over-paths" picture of quantum mechanics. As you can see, though, it's just a different way of looking at exactly the same math, namely multiplication of matrices and vectors.

So then why use the sum-over-paths picture at all? Well, a few reasons: physicists like it because it often gives them more insight into what's going on, into what are the more and less important contributions to a given process, and it can also make calculations easier. Meanwhile, computer scientists like the picture because it lets us simulate a quantum computer by a classical computer, still using exponential time but now using only a linear amount of memory, rather than the exponential amount of memory we'd need if we tried to store all 2^n amplitudes at once.