Carbon-14 is only good for dating things that are less than 50,000 years old, due to it's relatively short half-life. Scientists are also well aware of the degree to which carbon concentration in the atmosphere can vary depending upon the geological period, as well as the processes which govern it (the carbon cycle, volcanic activity, carbon weathering, etc), and how to determine the concentration (radiocarbon dating and dendrochronology).
Even if everything we knew about radiocarbon dating was wrong (it isn't), it would do absolutely nothing to transform our current understanding of the age of the Earth. We have other methods (such as uranium-lead dating) which are far more accurate for that.
Relativity states that the faster you travel, time slows down, but the kind of speeds an object would need to be travelling in order to experience a significant difference in time dilation are so large that it's not especially relevant to calculations concerning the age of the universe (an object would need to be travelling roughly 87% the speed of light just to slow time down by 50%, for example). It's also known that cosmic expansion is accelerating, which would suggest that time would have moved faster, not slower, during the early universe than it does now.
How we calculate cosmic expansion is by looking at the observable universe and calculating how fast galaxies are moving away from one another (typically by monitoring how redshifted they are). The absolute size of the universe is not especially relevant, since cosmic expansion works in all directions everywhere, and is a property of space itself. What's relevant is knowing how densely concentrated all of the matter and energy in the observable universe would have been at arbitrary intervals into the past, and it's here how we work out when the big bang happened.