"How are things going?" I'm doing well.

I'm going to go with "not something worth giving two shyts about" or maybe some phrase like a "waste of keystrokes".

Is the other person a native English speaker? their responses are colloquially correct, if not grammatically correct.

I think the word you are looking for is "colloquial". When two people are communicating they are expressing a feeling or an idea. The mathematical correctness of the expression isn't important.

I don't think either of these are grammatically incorrect.

In the first example "how are things going" is an idiomatic expression that is used as part of a social ritual. The response is another idiomatic expression. This is a typical greeting ritual.

The phrase "A big nerd is just who I am" is perfectly correct in English. When Darth Vader says "Luke, I am your father", it is the same sort of thing.