No, "to shop" in Norwegian would be "å handle" (å handle - handler - handlet - har handlet), or the more recent "å shoppe" (å shoppe - shopper - shoppet - har shoppet). "Å kjøpe" means the specific action of buying something.
Å kjøpe: Jeg kjøper en TV (I'm buying a TV) Å shoppe: Jeg shopper for mye (I shop too much)
It's neither. Look at this page and listen to the sound file. It's the "ch" sound in the German word "ich". https://en.wikipedia.org/wiki/Voiceless_palatal_fricative