The universe did not come from nothing. Consider the following two claims:
* The universe has a finite age.
* The universe has always existed.
Both of these claims are true. It might seem that if something has always existed, it must have existed for infinitely long. No so. At every time in history, the universe has existed, even though the age is finite.
Your intuition might disagree. But recognize your intuition has been developed under very peculiar circumstances on the surface of the Earth. You should be skeptical of your own intuition when studying things outside of your usual experience. In this case, human intuition says two things:
* There was always something before.
* There was always something 5 minutes before.
In ordinary human life, both claims are true. That's what your intuition says. But for the universe as a whole, only the first claim is true, and the second claim is false. Even though it is true that there was always something "before" (for any time you give me, I can divide that time in two and give you an earlier time), it is nevertheless not true that there was always something 5 minutes before (pick 4 minutes after the big bang, and 5 minutes earlier than that is simply undefined [1]).
Big Bang theory is a self-consistent, meaningful mathematical theory. It does not fly in the face of logical reasoning. Rather, it is a spectacular case-in-point of the power of logical reasoning to describe and explain the universe.
Your second point is that the Big Bang disobeys the first law of thermodynamics, which is the conservation of energy. False. You are assuming there was a period of time when there was no energy, and then suddenly energy came into being. That's not what happened. The energy has always been around, for all time, and thus the first law is not violated.
[1] It so happens that "negative time", or even time t=0, is equally as meaningless as Elephant Time or Verb Time. They are all undefined.