Why the "lake"?
Jina is a Ji/Ma noun, and it's possessive pronouns use la-(sing) and ya-(pl) as prefixes.
I think the question was more: why is it jina lake la mwalimu instead of just jina la mwalimu. I think it's just a stylistic preference, but I'm not sure.
Is it at all correct to say "Jina lake la mwalimu"?
Why is it nani and not nini?
I understand that
Who is his teacher would be
Mwalimu wake ni nani?
But why is
what is your name
Jina lako ni nani?
when do you use nini?
Nini is for things, not people.
Mwalimu wake, jina lake nani - should be accepted