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.
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.
The historical evolution of diagonal argument from set theory to computability and logic is described in:
The Diagonal Argument - a study of cases, International Studies in the Philosophy of Science 6, 191-203 (1992).
The diagonal argument as a uniform principle for generation of set theoretical paradoxes is introduced in:
Cantor's theorem and paradoxical classes, Zeitschr. f. math. Logik und Grundlagen d. Math. 32, 221-226 (1986).
