Do you need a purpose to exist?
Yes I do. Life without a purpose is boring and pointless. Having a list objectives is something good in life because it also motivates your actions.

Do you need a reason to live?
The ultimate reason to live is survival.

Do people need to know who you are for you to exist?
No but I might get frustrated.

Can you live with being a nobody?
I value privacy a lot but I appreciate any form of recognition.

Can you exist with being ignored by the masses?
Surely, just not by friends.

Do you need recognition to think you even reside in this plane of existence?
I don't understand what you mean.

If you weren't recognized by anyone, would you cease to exist?
No because in that case I'm not the problem. It's them.

If your purpose was fulfilled, would you continue to live?
Yes, I would help other people with their own purposes.

Would you even have a reason to?
Already answered.