Is the universe infinitely old?

Yes indeed, the universe is infinitely old and of infinite extent. Current cosmogony relies, instead, on the interpretation of the galactic redshift as purely the result of the Doppler Effect, thus leading to the popular idea that the universe is expanding. However, when light is considered as wave motion within a medium it becomes clear that the redshift is also produced by absorption over distance, as in other types of wave motion. In fact, even the Doppler Effect requires wave motion in a medium and could not occur in the empty space proposed by Einstein (an idea he himself rejected in 1920, with much of the "scientific" community still to catch up).

The best summary of Infinite Universe Theory is at:

http://scientificphilosophy.com/Downloads/IUT.pdf

To understand its foundation, read "The Ten Assumptions of Science" or Chapter 3 in "The Scientific Worldview."