Çalmak also means vurmak "to hit", as you can see in the sentence "al da başına çal"
Also there is another meaning of çalmak let me explain it with and example: "onun gözlerinin rengi yeşile çalıyor" means color of his eyes is not exactly green but something similar to green 
So there are many different meanings of this word. I think you are right there must be connection between them but I don´t know what it might be.
I also think it is not coincidental: one syllable verb roots hardly just lapse together.
One possible explanation is foreign influence. The corresponding verb in Arabic or Persian may have a range of meanings and these may have been transferred from language to language by bilingual speakers.
