Senin kalemim var mı? 

(My attempt - senin is ´yours´, kalemim is ´my pencil´ but it doesn´t make sense as a sentence)

Hayır, benim kalemim yok.

(My attempt - Hayır is ´No´, benim is ´mine´, kalemim is ´my pencil´.  So should the sentence read as "No, this is not my pencil"?)

 it is confused me too

i think it is wrong , so the right is  Senin kalemin var mı(is  this your pencil ?

so the answer be

Hayır, benim kalemim yok. (this is not my pencil )

I think it actually means ´do you have a pen?´  ´no, I don´t have a pen´

 Agree with:

1. You.  The original sentence is wrong.  It must be senin kalemin var mı?

2. Sonunda.  For the translations to English.

Is this your pencil would be Bu kalem senin mi?

This is not my pencil would be Hayir, benim kalemim değil.

hayir, benim kalemim yok actually means no, i dont have a pencil.

turkish logics make me craaaaaaaaaazy!