Does the age of the universe take into account General Relativity / Special Relativity? It is generally accepted that the age of the universe is approximately 12-15 billion years old based on the speed of the expansion of the universe. Since everything is moving very fast away from us, doesn't relativistic effects from high speeds come into play here? If something is moving quite fast, time slows down, affecting how "old" it is. Is all this taken into account when calculating the age of the universe? I don't know if I'm making any sense to anyone else or not.
This is a confusing question - your title mentions GR, but of course the age of the universe is entirely derived as a result of using GR to solve for the dynamics of the expanding universe.

The text of your question is talking about time dilation and the effects of special relativity (a subset of GR). Here there is a point to be addressed. It is not a good way to think about the explansion of the universe in terms of things receding from each other at ever increasing velocities. It is better to think about space expanding. It is also not possible to apply special relativity except in local inertial frames.

It can and has been shown that the redshift-distance relationship cannot be simply interpreted as the doppler shift due to things moving away from us at ever increasing speeds - since this of course predicts a maximum recession velocity that approaches the speed of light. In fact, beyond redshifts of $z=1.5$, the "recession velocity" in all viable models exceeds the speed of light. What actually happens is that light emitted at some earlier epoch travels across expanding space to get to us and this expansion increases the wavelength of the light.

Essential reading: Davis &amp; Lineweaver (2003).