I'm not sure, but when I last flew out of JFK with SQ, I was told to use the Swiss Air lounge. Unless OZ contracted another lounge, chances are it's Swiss Air for OZ as well. The lounge was absolutely horrible and not worth visiting, let alone visiting early. Worst of all, the lounge was before security. That terminal of JFK is being renovated though, and if I read correctly elsewhere here it has since moved behind security.

The lounge in ICN is spacious and quite a pleasant place to be. There are various seating options depending on whether you want to eat, read, rest, etc. They have a room where you can use a laptop should you not have your own ready and even have several relaxation rooms with massage chairs (which are often already occupied though). Food, however, is no strong point. Warm food is only served at specific times and even then it's nothing to write home about.

I'm afraid I know nothing about HKT.