Yes. It means the same thing as “doing really well”, which is probably the more correct way to say it. However, you’ll often hear people say “doing good” in everyday speech.

You haven't given us any context, but it might be being used to mean "It looks like you've been making a lot of money." Perhaps they're saying that because you're driving a new car or wearing expensive clothes. And yes, it really should be "doing well" for it to be grammatically correct. "Good" can be an adjective or a noun, but it can't an adverb.

I think in this context "doing good" means the person seems well. It can be a very general kind of politeness, but I think here it suggests a person looks better than before, perhaps because of an improvement in health. Others have suggested that this phrase could indicate that the person appears to be doing well for himself as in wearing nice clothes---doing well financially. I believe in English to indicate what we would almost certainly use the phrase "doing well for yourself".

Yes, in the context of the sentence "you look like you're doing really good," "doing good" means "doing well." The phrase "doing good" can have different meanings depending on the context. In the context of performing acts of kindness or charitable work, "doing good" means engaging in activities that have a positive impact on others or society. However, in the context of the sentence you provided, it means that the person looks like they are succeeding or making progress in whatever they are doing.