So I thought about phrase "I'm 17 years old and I'm already having existential crisis" - is it correct to use "I'm already having" on the scale of 17 years? Because it is a pretty long period of time.
If talking about existential crisis, then it's implied that I had it a week ago, I have it today, may be I'm having it at the moment (like a panic attack) and I will have it in the nearest future.
Yes, your sentence is perfect the way it is. It's fine to use "already having" even though you are only 17 because this accurately describes your experience, and includes the nuance that you think 17 is not the right age for an existential crisis.
It's a little unclear to me whether you think 17 is too old or too young for this -- it seems like you think 17 is too old? But this is just normal ambiguity and not due to any grammatical errors.
