Light can only travel at one speed (in a vacuum), approximately 300,000 km/s. It doesn't matter what frame of reference it is created in, it never goes faster or slower than this speed, and it doesn't matter what frame of reference you are measuring it from, you will always measure it to be the same speed.
This is given by Maxwell's equations, Einstein's Theory of Relativity and backed up by all experiments we have done to test this. I don't think anything contradicts this including Julian Barbour's theory.
So in this sense the speed of light is special. In another sense the speed of light has to be something so whatever figure it is is not remarkable in itself, it is simply another constant of physics.
From what I can gauge from that article, Julian Barbour's theories are essentially the same as Einsteins theory of relativity in that they predict the same things. Where they differ seems to be in the separation of time from space-time and in the definition of a theory of gravity which is not based on space-time. In Julian Barbour's theory time is emergent (not tided to space) but otherwise roughly the same (e.g. time dilation still occurs). This had some repercussions for gravity at larger distances (due to the different interpretation of time) that may effect our understanding of dark matter and dark energy. But nothing conclusive so far.
Another "absolute" in Einstein's relativity, which the article did not go over too much, is light. Einstein thought light is the "speed limit" of the universe. The way I understand it, any 2 people, no matter how far apart they are or how fast they are going, see an event at a 3rd location as happening at the same time, relative to their locations.
Your understanding is incorrect. You cannot compare clocks in different frames of reference, nor agree on an event in a third frame of reference as having occurred at the same time in those two frames of reference. You are correct that the speed of light seems to be the 'speed limit' of the universe. All experiments we have done bear this out (nothing has ever been measured going faster than the speed of light) and it seems likely that nothing can be. I couldn't see anywhere that Barbour's theory differed from or contradicted this.
Therefore, whenever you move, you travel through not just space, but also time (called time dilation), because others must see you at each different place at the same time as each other.
We are always travelling through time (time always travels forwards) and essentially Earth is moving through space so we are always moving. I don't think this the right way to look at things. Its better to think of time dilation as occurring as speeds approach the speed of light. Time dilation simply means that time slows down for that frame of reference (relative to a stationary one). Both frames are still there and can measure things, its just that their clocks may differ in the times they record for events outside their frame of reference.
Therefore, we can not travel at the speed of light, because then others would see us at all places at once.
No this is wrong. We can't travel at the speed of light as it requires increasingly more energy to accelerate us to that speed (infinitely so) so we can never reach it. But extrapolating (thought experiment only) then if we see someone in a spaceship travelling past us at the speed of light, we see time frozen for them (e.g. no movement whatsoever in the space ship, including no aging of the pilot, no electrical signals, no movement of air particles, etc). The spaceship is entirely frozen relative to itself, but relative to us it is flying by at the speed of light. It's not everywhere, its still a single entity moving past.
I believe that if we go faster than the speed of light, we travel backwards in time because others see us going backwards. If any of this is wrong or confusing (I'm sure it is), please feel free to ask for clarification or just edit if you have enough rep.
People have postulated this but currently it is just extrapolating the laws of physics past their known limits. As far as we know you can't go faster than the speed of light so this can't occur.
Is the speed of light special in Julian Barbour's theories of relativity like it is in Einstein's?
I'm just going by what I read in the link to the article you provided but from what I understand Barbour's theory doesn't treat the speed of light any differently from Einstein's theory of relativity. E.g. its still the speed limit for all matter and nothing can go past it. His theory does separate time from space, but I don't think this leads to any new behaviors of time (e.g. travelling backwards in time) or for time dilation at relativistic speeds (e.g. it would still behave like Einsteins models predict).