The soundtrack to the 1st series was sublime, with most of it being of the era... with a few exceptions. For example Peter Gabriel's beautiful cover of Heroes was recorded in 2010, but with it being a 1970s song, and Gabriel's voice being a defining one of the time, it worked well with the show.
Atmosphere by Joy Division (1980), Should I Stay or Should I go by The Clash (1981), Nocturnal Me by Echo and the Bunnymen (1984), Elegia by New Order (1985) - so not strictly 1983, but it very much captured the feel of the time.
Regarding goth. The term was first used (I believe) by a journalist to describe one of the Joy Division albums, and the fashion very much developed from the likes of Bauhaus, Siouxsie Sioux and Robert Smith in the early 1980s here in the UK, before spreading across the water.... The Cure hit the big time here in about 1981/2 with their Faith and Pornography albums selling well, and first got a top 10 hit single in 1983 (The Love Cats).
Source - me

. It was, and still is, my music genre of choice... and I sat in a famous black chair on a British TV quiz show and answered questions on Joy Division.