You are absolutely correct! It's hard enough that there are so many rules, and that each rule has exceptions. Despite all the rules and formal exceptions, expect native English speakers to break (almost) all the rules in casual conversation.

Grammatically correct: "She is doing well." Often used: "She is doing good." Many Americans have a terrible sense of grammar!

She is doing well is the right way. Doing is a verb . It can easily take 'well' which is an adverb.

You are right. Well is more formal.

Both are correct.