1) Nope. Imagine I randomly select a ball from a set of a blue ball and a red ball. I send one to A and one to B. When A gets his ball, he'll immediately know which one B got. Can A send information 'instantly' to B this way? No.
Entanglement is weirder than that, because the way quantum mechanics works, it's not a case of merely not-knowing if the ball is red or blue; the balls are in a truly undefined state until their color is 'measured', entangled such that one must be blue and the other then red, but without actually having any real but 'hidden' value. So this raises the strange question of how the ball at A 'knows' it's supposed to be blue once the ball at B has been measured to be red? It appears as if they're somehow 'communicating' (although it's not necessarily the case), and that communication (if it exists) would be instantaneous.
But from the standpoint of the people at A and B this makes no difference. They still can't use the entanglement for communication.
2) As for tunneling, it occurs instantaneously. But what tunneling is, is having a particle in a similarly undefined location, where there's probability of finding the particle in one region or another, even though there may be an exactly zero probability of finding the particle in the areas in-between. You can't really say the particle 'traveled' from point A to point B, since by definition, it doesn't have a location, just a probability of being in a location.
Once a particle has been measured at a particular location, though, this probability distribution evolves at a finite speed. So once it's been measured at point A, you will - to an extremely high degree of probability - not be able to measure it at B until after some time that doesn't exceed the speed of light.