The key to travelling faster than the speed of light lies in accelleration. To ACCELERATE to the speed of light would require infinite energy. However to travel at the speed of light does not inherently require any energy. Additionally, relativistic time dilation occurs during acceleration, not because of velocity. Thus, if there were some way to achieve the speed of light without accelerating to it (i.e. to simply change state such that you instantly travel the speed of light without accelerating) then that would bypass all of the problems with relativity. How that can be accomplished is beyond my understanding however there are events in nature which can cause this. Hawking Radiation is such a thing. Hawking Particles are particles which are capable of escaping a Black Hole. As we know this is only possible if they are travelling faster than the speed of light. They are. Additionally, they have mass so aparently it is possible to achieve speeds faster than light. However Hawking particles do not accellerate to this speed; they simply achieve it. It has to do with particle pairs and Quantum Theory. According to Quantum Theory, particle pairs "appear" all the time. However they always come in matter/anti-matter opposites. Thus the moment they appear they are cancelled out. However, when such a pair appears such that one particle is inside the event horizon of a black hole, and the other is on the outside, the pair is split; the one inside remains inside, while the other shoots off at faster than light speed. This is also how black holes manage to "bleed" off mass constantly over time, since the mass that was lost came from the black hole. It is also why black holes can become unstabel and eventually explode. Anyway, faster than light travel is possible. Acceleration to light speed or beyond is impossible.
As for time travel, my belief is that were one to travel into the past, they would not be capable of travelling into their own past. The past they travel to would of course be an alternate reality, even if they did nothing in it. Thus they would be free to do whatever they want without causing any kind of paradox. If they killed their younger selves, they would continue to exist because their origin is from an entirely different reality to begin with. Then when they travel back into the future, they are once again travelling to an alternate reality, but in this instance one more closely related to the reality you were just in. Thus, the future you travel to would be affected by the events the time traveller kicked off in the past.