The reason people give the responses you label as “wrong” is because well and good have more meanings than the ones you cite.
Merriam-Webster, for example, gives definitions that make sense of the three examples you label “wrong”:
For “I’m well”, there is well (adjective) 1b: being in satisfactory condition or circumstances
For “I’m good”, there is good (adjective) 2e: free from infirmity or sorrow
For “I’m doing good”, there is good (adverb) 1: WELL
So they all seem fine to me.
As Cerberus wrote about 'U' English, replying to "How do you do" with "How do you do" used to be the case among some classes in England (at least), but it seems to be (sadly) nearly extinct. Kate Fox writes in the first chapter of Watching the English (which is about talking about the weather):
We used to have another option, at least for some social situations, but the ‘How do you do?’ greeting (to which the apparently ludicrous correct response is to repeat the question back ‘How do you do?’) is now regarded by many as somewhat archaic, and is no longer the universal standard greeting. The ‘Nice day, isn’t it?’ exchange must, however, be understood in the same light, and not taken literally: ‘How do you do?’ is not a real question about health or well-being, and ‘Nice day, isn’t it?’ is not a real question about the weather.
So in this usage, "How do you do?" wasn't an actual question about the person: it was just a meaningless greeting, and for instance what one might say when introduced to someone (in lieu of "Pleased to meet you"), presumably while tipping one's hat. In reply, the other person, also wishing to make the same meaningless greeting, would say "How do you do".
Note that "How do you do" in this usage was even spoken as a statement, not as a question (i.e., without a rising tone at the end).
These days "How do you do?" is more likely to be interpreted as an actual question. Interestingly, as n0nChun observes, one does sometimes hear a similar exchange these days, with "What's up" getting the reply "What's up", or even just "sup" — "sup".
Best Answer
"How is doing?" is definitely not an acceptable greeting in any form of English with which I am familiar. Most likely you've heard an extremely reduced form of "How are you doing?" The levels of reduction go like this:
How are you doing?
How you doing?
How ya doin'? [hajə'duɪn]
It's also possible that some people said "How's ya doing?", as the word how's (also sometimes spelled howse) is a variant of how in some dialects.