The universe didn't start with one single spot, it was just more and more dense the earlier and earlier you go. A singularity is a mathematical artifact that tells you your theory isn't up to the task of describing your phenomenon.
And nothing can
accelerate to and past the speed of light, but things "move" faster than light all the time, depending on your definition of what "move" is. To move, you need a set of coordinates, and that's something that is very different in GR than it is to our normal idea of Cartesian coordinates. See
Comoving coordinates.
There are galaxies that are receding away from us at faster than the speed of light right now, but they aren't moving through space faster than light in their own frame of reference. Space is expanding so the galaxy is being carried away from us faster than light.
Think of it this way. Shine one flashlight in one direction, and another in another. To you, the distance between the "front" of the beams of light is increasing at 2 times the speed of light, and that's ok.