The most famous hotdog stand in Reykjavik is Bæjarins Bestu (means "Best in Town") on Tryggvagata across the street from SALT restaurant at the Radisson 1919 hotel.
The best restaurant in Reykjavik at the moment is called Seafood Cellar in English http://www.sjavarkjallarinn.is/englis... It's Asian-influenced using many local ingredients.
For more traditional Icelandic seafood and lamb, try Siggi Hall, Tveir Fiskar, Thrir Frakkar or Humarhusid (Lobster House)
I've never had a bad hot dog in Iceland. PULSAR! is the word to look for . . .
Get it with everything - the mustardymayo and crispy onions. You can buy hot dogs (and frozen leg of lamb) at the airport to take home --they are a legal declarable import to the USA.
The hot dog stand at the bus station is legendary, but really, the truck outside the regional swimming pool, or the tiny one tucked in one of those shopping streets leading down to the main plaza are all fab.
I would avoid Asian cuisine in Iceland at all costs, unless something has changed in the past year. We were looking for something a little different after a long trip in Europe and the Thai food we tried was practically inedible.
I love Iceland -- been several times over the years. Have a great time!