Recently, I finished my phone job interview with the phrase "Thanks for having me". It was a reply to the other person's "Thank you for your time". So, does "thanks for having me" sound alright in this context?
No. "Having me" implies "having me over", usually in the sense of for a visit or stay, so wouldn't be right in a professional context even if face to face.
I would suggest a response of simply "thank you" to "thanks for your time" as an appropriate professional alternative that will also work on the phone.
Not really; I'd use this as a polite farewell phrase on leaving a dinner party or having visited friends for the weekend; see here. I wouldn't think it quite in order for a business arrangement; perhaps a simple no trouble, thank you would be better.