"The cell phone is old."
Yes all 3 of those are used. You'll also hear 전화기 (my personal favorite).
Yup. Implementation details don't belong in everyday vocabulary unless it serves to distinguish things that could be confused in common contexts. I mean, I would never want to waste my breath to painstakingly tell my friends to pass me my mobile phone, nor would I call tech support to specify that my Microsoft Windows Internet Explorer is not working.