Why 13.7 billion years

The current best estimate of the time since the big bang is 13.75 +/- 0.11 billion years. That value is based on several studies, including mainly the Wilkinson Microwave Anisotropy Probe. Otherwise, I don't understand your question. The +/- 0.11 billion years means that this is not an "exact" estimate.

Also, it takes about that long for three generations of stars to happen, for the solar system to form, and life and then humans to evolve.

The universe did not spawn out of nothing. That is perhaps the most common misconception about the Big Bang. Most people don't seem to understand that "nothing" in physics/cosmology is not the same as absolute nothingness. There was something that started the Big Bang, but that something is unknown.

Also, the total energy in the universe is not 0, if it was there would be absolutely nothing. You may be referring to the net energy, but when we talk about the shape of the universe being flat we're talking about the ratio of vacuum and matter densities of the universe and that has nothing to do with net energy.

Also, none of this really has anything to do with the age of the universe, the age of the universe is determined by the redshift in the galaxies among other methods. Redshift is actually predicted by the Inflation Theory model and it has been verified. The only way for redshift of galaxies to occur is for space itself to be expanding since light doesn't travel through any particular medium. Redshift observations show us that the farthest galaxies are 13.7 billion light years away and this tells us that the light from them took approximately 13.7 billion years to reach us. This is under the assumption that the speed of light is constant, however, that's not just an assumption, you run into all sorts of problems with physics if you change the speed of light. I won't go into all of that though, but just know that the implications of fluctuating light speed also change the fundamental principles governing motion and matter.