"Good" is an adjective, i.e. it modifies a noun. "Well" is an adverb, i.e. it modies a verb. Adding some confusion, "well" can also be an adjective meaning "healthy". "Good" can also be a noun, meaning something that is morally right or beneficial to society, as in "Giving to the poor is a great good."
If a sentence uses "good" as an adverb, it is a grammar error. It is a very common error in English, but I would still call it an error and not an idiom.
(As has been discussed on this forum in the past, language is largely or, some would say, entirely defined by what people commonly say, so if an error becomes very common, at some point we must say that it is now the standard. But I don't think using "good" as an adverb has reached that point.)
In some cases the structure of a sentence is such that more than one part of speech would make grammatical sense, though normally giving a different meaning. Your examples fall into either this category or the multiple meanings of "well" and "good" I mentioned a moment ago.
So in your examples:
"I did well" -> "well" is an adverb -> I completed the task correctly.
"I did good" -> "good" is a noun -> I performed a morally right action. The opposite of "I did evil." If the writer means that he performed the task correctly, this is an error and it should be "well".
"I feel well" -> "well" as an adjective -> I am healthy. In context, it could conceivably mean that I perform well at the task of feeling, like "I see well" meaning I have good eyesight, but this seems a rather unlikely thing to say.
"I feel good" -> "good" as an adjective -> I am satisfied with my present condition. Could mean healthy in which case the meaning would be essentially the same as "I feel well". The construction is the same as "I feel happy" or "I feel cold."
"I am doing well" -> "well" as an adverb -> Depending on context, either I am performing tasks correctly -- "Are you getting the job done?" "Yes, I am doing well"; or my personal state is good -- "Are you recovering from your illness?" "Yes, I am doing well".
"I am doing good" -> "good" as a noun -> I am doing things that are beneficial -> "Why are you volunteering at the homeless shelter?" "Why, I am doing good."
"I will be good" -> "good" as an adjective -> I will behave properly. "Billy, will you behave for the babysitter?" "Yes, mommy. I will be good." Depending on context, it could also mean that I will be in a desirable state. Like, "Oh, I didn't make enough food for dinner." "Don't worry, it's enough. I will be good."
"I will be well" -> "well" as an adjective -> I will be healthy.