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All these three theorems (I am not 100% sure about the third, but I have heard it has a similar argument with the other two) use self-referentiability as contradiction and they talk about the impossibility to solve everything.

Is there anything that generalizes this concept? I think they actually describe the same object.

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Yes: see Lawvere's fixed-point theorem.

There is an expository paper by Yanofsky covering exactly the results you mentioned (and more): A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points.

See also this MathOverflow question.

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