[harmoat]

Since these 3 results are so similar are they a subset of a more general concept?

[moomin]

Arguably the halting problem is just a subset of Godel where every theorem is "This does or does not halt". Someone's written a paper going the other way: https://www.andrew.cmu.edu/user/avigad/Teaching/halting.pdf

The crucial trick is that you can plug in an entry as data to another "program/theorem". This is extremely interesting, because there are limited logics and limited computation models where Godel and the Halting Problem do not apply.

Not so sure about the diagonal argument.