No never, if someone said “me doing good too” it would sound like a child talking, the only time you would say me at the start of the sentence would be like a question. How are you doing? Me? Oh I’m good thanks

"Me doing good too" is not correct grammatically even if one was using it as colloquial English. But . . . There always seems to be a "but" with languages! If you were clowning around 🤡 with friends or family you might say it as a form of over exaggerated grammatical error. In other words, an "on purpose" mistake. So, it's not proper to use it as informal or, of course, formal speech.

British English and American English native speakers would never say, 'Me doing good, too.'

B

Me doing good is understandable, but rather informal to my ears. You can simply say " Me ,too."