If you make the supposition that no thing inside the universe could generate the universe and that every thing that exists is inside the universe than the direct conclusion is that the universe was not generated by a thing, which is similar to saying that it was generated by nothing.
The difficulty here is that you have to ask yourself what you mean by "thing" and "nothing" and whether "nothing" means the same thing as "not thing". The field of philosophy has historically been very interested in questions of this nature.
There are several places that this argument can be attacked. The first would be to attack the premise, and argue that there might be entities outside of the space time continuum. Platonic forms would be an example of proposed entities that exist but are not "things" inside the universe.
We could also attack the application of cause--saying that the universe "came from" something implies rules of cause and effect that only exist inside the universe itself. The universe didn't "come from" nothing, because the words "come from" have no meaning outside of the universe. That would be the scientific position... that you can't use concepts relating to time and space outside of time and space.
Based on the former point, you can't assume that a universe can't create itself. Since time, space, and the laws of cause and effect don't apply outside the universe, some form of cosmic event could be occurring inside the universe that somehow reaches outside the universe and in fact is what creates the universe. So premise 2 could be invalid.
There is another flaw in this proof, which is that it defines the universe--that is, the space time continuum itself--as a thing bounded by the same rules as every other thing in the universe. However, there are properties of the universe which are not properties of any thing within the universe, for example, the universe itself can expand (that is, generate additional space and time) without apparently getting that space and time "from" anywhere. So it isn't clear that the universe is a "thing" in the sense that a particle is a "thing". It all comes down to definitions.
Based on comments, you seem to believe that we cannot reject 11 without rejecting 2. That isn't true. As I've pointed out, we can reject 1 or 2. Most people reject 1, that is, the majority of humans believe that there is more to reality than the universe of things, and given the nature of the universe, 2 is not supportable when applied to the universe since cause and effect are only valid within the universe. We could point out that since anti-matter exists and moves backwards in time, 2 is invalid even within the universe.
Further, we could accept 1 and 2 and still reject your argument by questioning whether the universe is itself a "thing". This is a major unstated premise of your argument, and without it your argument does not hold at all. In fact, it would be just as valid to use your reasoning to show that the universe is not a thing! If you insist that the universe is a thing, we could point out the difficulty of having a set contain itself, since this leads to Russel's paradox, and question what version of set theory you are using and attack that.
The fact is, it's difficult to prove things about the origin of the universe. That's why there is a field of cosmology at all, and why the field of philosophy continues to have new ideas despite having thousands of years of literature to draw on.