"time running slower for things moving at an appreciable proportion of c" - Time doesn't slow down, perception of time changes from the observers point of view.
Mercury's orbit is easily predicted with newtonian laws, again, perception of it changes due to the strong gravitational effect of the sun on it and its light we use to measure it.
In other words, I really don't accept things like time travel, infinite mass, getting younger if traveling at the speed of light and bullshit like that.
I don't know if they use that stuff to make it interesting to the unwashed masses but it is an insult to our intellect.
Going back to the speed of light as a measure of energy, I take as you say, it is the speed of all electromagnetic radiation, but why related to energy and mass?
And what if there is something faster? say like hypothetical tachyons?