They can use the fact it was released in November 1983 in Canada before the U.S. in January 1984. Steve's car radio was picking up Canadian radio stations. Timeframe works since a month later it's Christmas.

Also the song continues playing while Steve is running through the yard and still continues to play while he is climbing up to Nancy's window. But I suppose Ted could of had speakers installed in the trees in the yard and on the outside of the house and just so happen to be broadcasting the same song at the exact same timethen yes Steve could hear the song while he's climbing up the house.

One of these days I'm going to cut you into little pieces.

Unclear, but as there wasn't any noticeable Doppler effect on the song as the car drove by you could argue it was non-diegetic, so no.