Were shall I start?

With the obious: you do not understand the wikipedia citation. The statement there does not start, end or use within anything like an "assumption" that 9 = -9.

g presents to f an indecidable problem as it reference the output of f and turns that into its negation.

If f says g stops, g will run forever and if f says it will run forever g will stop. So the output of f is always wrong. If f would attempt to solve the halt problem on g by executing it, it would run into an infinite iteration of g -> f -> g -> f ....

The proof is very similar to the proof Goedel provided for the necessary existance of undecidable statements in general. g is an example of the pattern "This statement is false."

f(g) makes as much sense as f on any other function. As f claims to be the function to solve the halt problem on ANY function. g is in that set. As f can't solve the halt problem on g, f's claim is rejected.

If you do not understand the purpose of knowing that, don't worry. It is not relevant from an engineering perspective, no one will ever engineer anything like g. But it is very relevant for math and information science.

And BTW Turing machines do have infinite memory by definition.

This is science, boy, and we both know how bad you are at that.