Is it common to describe one’s life stage in English, like “Stage III Hemingways”?

This is figurative usage. I would not say it's common, but I believe it would be widely understood in the US.

Just as you associate Stage III with a medical term indicating that you have a "serious case" of something, if you poetically call someone a stage-three X, it's as if that X is a disease, and the person in question has got it bad.

What's important to recognize about this phrase is that it's generally used in a mocking way. In this context, the narrator is mocking overweight older men who have dreams of being like the young, outdoorsy, manly author Hemingway.

Another example: If your best friend went away to college and became obsessed with indie music, oversized glasses and tight pants, when you met on winter break you might say she'd become a stage-three hipster.

Probably because the phrase is intended to insult or mock someone, you don't really hear about stage-two Xs - why insult people less when you can insult them more? (But there's no reason you couldn't use it - e.g., If Max is a stage-three band geek, Mary is just a stage-two.)

You would probably not use this to describe Obama (unless it were as part of an insult similar to the ones above).