There’s a big, big difference between a “normal speed” and “relativistic speeds” when you approach (but don’t exceed) lightspeed, though. I wouldn’t consider .95c to be anything close to a “normal speed,” and that’s still only 95% the speed of light. In my mind, you have to hit a relativistic speed like .95c to punch a hole through realspace into hyperspace, which allows you to travel faster than c (aka lightspeed).
No, when something speeds up, its mass increases compared with its mass at rest. As the speed of an object increases and starts to reach appreciable fractions of the speed of light, the portion of energy going into making the object more massive gets bigger and bigger. This explains why nothing can travel faster than light (using normal motion) – at or near light speed, any extra energy you put into an object does not make it move faster but just increases its mass.