The universe cannot be infinitely old or all useable energy would have been lost already (entropy). This has not occurred. Therefore, the universe is not infinitely old.
You're making an assumption about the nature of the universe.

The second law of thermodynamics states that the entropy of an isolated system tends to a maximum. Usually the universe is assumed to be an isolated thermodynamic system, but this is a circumstantially ratified assumption and carries no connotation of "correct."

So it could also be assumed, just as rationally, that the universe is not thermodynamically isolated, and by the second law, the entropy of a system that isn't isolated may decrease.

The universe could be infinitely old and be doing nothing but gaining usable energy.

It [the universe] could not be eternal since that would mean that an infinite amount of time had to be crossed to get to the present.
Again, you're making assumption. You've got an image of time as a flat linear form, with the possibility of "infinity" in either direction of the line.

Yet you claim that it can be "crossed," which infers that you believe it to have dimensional form. Any shape with dimensional form (of any order of magnitude) cannot be infinite. It must have borders, and if a higher dimensional form upon which the shape in question lies is posited to exist, then it must have limits. Time is referred to as a fourth dimensional shape, not a "line" by which we measure the order of our three dimensional movements, so it must have a border and a fifth dimensional plane upon which to lie.