Astrophysics -state of the universe

I am wondering if anyone has an opinion on this: Could we (our solar system) be traveling at the speed of light or close to it? It seems to me that if inflation is correct and the known universe is still expanding then we (pick a limit Earth, solar system, milky way etc.) must have traveled from the center to our current position which we should be able to estimate velocity from the estimated center of the universe. Given that anything gains mass and therefore gravity as it approaches the speed of light could our velocity account for the level of gravity and certain properties of elements and compounds here? Also could dark matter be slow moving, cold or stationary matter that has different properties due to it's lower velocity (how it interacts with light for example) and also how much gravity dark matter has. If you can run the numbers for the amount of matter we expect to find in the universe and adjust for velocity I wonder how it would come out? I would think there would be any easy way to come up with at least an estimate of our velocity estimating the distance traveled in about 14 billion years. Please let me know if anyone has any input on this question. Thank you.