I think you're missing the "modal nuances."
In modal logic, "necessary" means "true in all possible worlds." It doesn't mean "required to exist."
Thus, if you're talking about modal logic, necessity does
not entail actuality. For example, per Kripke it is necessary that the name "Harry Potter" necessarily refers to a fictional wizard created by JK Rowling. Names as rigid designators does not mean that the target of the reference is an actual human being.
Yes, it is.
At a bare minimum, you have to actually prove that the deity of your choice is necessary (in the sense of "required to exist") -- and a logical argument can't do that for you. It is not the job of logic to prove that a premise describes an actual state of affairs in the world.
I might add that I can certainly imagine a consistent universe existing without any input or direction whatsoever from any deity. I am not aware of any contradictions that result from positing a godless universe. In fact, I can think of many logical contradictions that result from including a deity, especially if said deity happens to be omnipotent and/or self-creating; I can also think of many ways it clashes with what we know of reality, such as how omniscience is incompatible with quantum mechanics.
So is defining God as "necessary."
In formal logic, a conditional is not the same as "if... then" in ordinary language. In formal logic, a conditional is true if the antecedent is true.
When a conditional is rendered into ordinary language:
"If God exists" is true, and "God must exist" is true, then the argument is true
"If God exists" is false, and "God must exist" is true, then the argument is false
"If God exists" is true, and "God must exist" is false, then the argument is true
"If God exists" is false, and "God must exist" is false, then the argument is false
Plus, you declared it to be a definition. Conditionals are not definitions.
Somehow, this claim seems rather unlikely.
At a minimum, you are obviously associating properties with this term "God," such as the necessity of its existence. That also means that "God" has whatever properties are required to make its existence necessary.
And again, logical proofs don't prove that things exist. They only verify that an argument is sound. To wit:
• All nudibranches are mammals.
• Mammals exist.
• Therefore, nudibranches exist.
The argument is sound, but this does not prove that "nudibranches exist" is an accurate description of the world, because nudibranches are actually molluscs. It's not the job of logic to prove that the premises are accurate.