In the city centre you will find it difficult to find a hotel in this price range. I would recommend as above the Cabin chain as above or Steel House ( this is basic and often sold out). A little above your price range 200-250 you'll probably have more luck. Any of the hotels in the Guldsmenden chain (Manon Les Suites is my personal favourite of these).

For local nightlife Versteboro or the City Centre. ( the latter more upscale options and the former if you're younger). You'll find Copenhagen is mostly a city of small cocktail bars. You can't go wrong with most of them listed in any 'best of' guide.

Tivoli Foodhall or Westmarket are both interesting takes on street food for a moderately priced meals.
Some favourite mid-priced "Danishy"restaurants are Pony in Vesterboro. Restraurant Gorilla, Marv & Ben's or Host closer to city centre.
If you feel like breaking the bank, you might get a last minute spot at the "New Norma" (although I personally prefer 108 or the Kong Hans in that range)