Do you know if that is a japanese adoption of "half" or is it based on pure Japanese etymology and just a coincidence it's sounds similar?
That one is a transliteration/adoption, as it is literally just ハ-フ (pronounced as "ha-fu"). The actual word for a half (like in "half a minute") would be 半 (pronounced as "han")
It's written using katakana, which denotes that it's a loanword.