I'm interpreting this question as being about "good" versus "well". "Good" is an adjective. You can evaluate a property or thing on a spectrum from bad to good, for example "this is a good example of how to use this word" or "this is a bad example of [...]." "Well" is an adverb. It's used to say something about how an action/activity is being performed. You could say that the thing is being done "poorly" or "well" or anything in between, for example "the student performed well in her test" or "you sang that very well." If you ask someone "How are you?", you'll *probably* hear "I'm good" in the US and "I'm well" in the UK. Justifying why those two answers (the first using an adjective and the second using an adverb) are both arguably correct requires some imagination. In the UK, "I'm good", taken out of context, is likely to be understood as "I am well-behaved". But in the US, the intention is "My health/mood/whatever is good". "There is some property or aspect of me that is good", hence the adjective is appropriate. As for "I'm well", the intent and implication is "I am doing well" (a robot might say "I am functioning well" :)). In those case you're judging the verb (doing/going/functioning) and so an adverb is appropriate. Your example is "how is it going?" So, strictly speaking the answer should be in the form of an adverb, so "well" is, on the face of it, what you'd expect. However, the US reply "Good" or "I'm good" or "It's going good" or Jerry's variants, are all to be expected, because of the implication that the respondent is talking about some property of themself. This is because the question is a little loose. What is "it"? When we hear that question we hear something like "How are things going" or "how is your life going" or "how are you". Hence, sentences structured around either "good" or "well" make sense, or can be made sense of if you understand the implications.

I'd say neither of them. I've never heard anyone respond to this question like that. You can say "I'm ok/fine." "Everything's ok/fine/alright." etc

#1 is the only one that works but it sounds a little odd. I don't use it. Pretty good, real good, not bad... these are acceptable positive responses in AmE. I suspect that a non-native speaker wrote this question.

You can say either, but "going well" is grammatically better. However, in the US many people use improper grammar depending on where they are and what type of culture or ethnicity they belong to. In fact, if you use proper grammar with certain natives, they may not understand you. I have this problem with a few co-workers who were born and raised in the USA.