The question is longstanding perplexing one. The difficulty starts right at the beginning, to properly assert the statement about "Nothingness", "Néant", "Nichts" etc.
The first difficulty is about "to be" and "to exist". Do they have the same meaning or not? First observe that from "X does not exist" it follows that "There is something that does not exist". Suppose that "to be" and "to exist" have the same meaning. Then we can say that from "X does not exist" it follows that "There exists something that does not exist" which is "somethings exists and something does not exist"! It seems that it is contradictory to say that "X does not exist" whatever X is! The argument can be put like this. To be able to attribute the predicate of "not exist" to X (or deprive X from the predicate of "exist"), X should have some form of "being".
To cope with this problem there are various approaches in the history of philosophy. The debate goes back to Sophist of Plato and continues up to this day. Some reject the rule of Existential Generalization, some make a difference between first order and second order predicate and also between "to be" and "to exist", some, like Quine, used Russell theory of description to deal with what is called "negative singular existential propositions". Some take "to exist" as a genuine predicate and some reject this. One cannot do justice to all of positions here.
Now I try to go over your arguments. But note that the problem here is about forming a meaningful proposition. What I tried to do was to show how these propositions are malformed and unclear and the problem is about the meaning of "to exist" and "to be". So the arguments I provide for the rest, are just immanent criticism of your arguments and might suffer from the same symptoms.
"Nothingness doesn't be, that's the definition of nothingness". The first question is about the meaning of what you say. According to what I said, are you saying "there exists something that does not exist". You have to say which property "Nothingness" possess or does not possess, when you say "it does not be". The second question is that can be we bring entities into "existence" just by positing them? I may define "multishape" as "an object which is both round and square and do not exist". Do I show it existence just by defining it?
"If there was nothing, nothingness would be, that is the only thing that there would be. Therefore, for nothingness not to be, something must be. Therefore, the existence of something is necessary." Same questions can be posed here. Here is the problem: Are the predicates "to exist" and "to be" similar the predicates like "to be red"?
Moreover is your first conclusion correct? From "There was nothing", you can say "There is no X such that X is", which is quit natural. If X ranges over all entities, including "Nothingness", then Nothingness is not. As simple as that. So you should say "nothingness would not be".
One way to circumvent the problem is to accept the difference between "to be" and "to exist" and then saying that "there are some objects that do not exist", like Sherlock Holmes for instance! Then "nothingness", whatever it is, does not exist. But then your argument for the existence of "Universe" will not work anymore.
Remark: The story can get even more complicated because some people accept the existence of sets but reject notions like nothingness as non-sensical.
Remark: A complete survey of answers to this question is indeed the history of philosophy in some sense.