Copenhagen in 3-4 Days

We were in Copenhagen a year ago, and started out in Hotel Alexandra. It's 2 short blocks from Tivoli Gardens and 4 blocks (walking) from the train station. They're showing online @ $122 per night right now. It's a very nice property.
When we returned from our cruise, we spent one night at Absalon Hotel--1 block from the train station. Its rooms have recently been completely redone, and the free breakfast is absolutely the best we've ever had anywhere in the world. Current rates are $166 on the online reservation companies.
When traveling to Scandinavia, you cannot get hung up too much on prices of hotels and meals. The whole region is deadly expensive versus most of the rest of the world, and you've just got to suck it up and pay the price. We found cruises the most reasonable way of traveling the region--where everything's included.
I suggest you get a good travel guide on Copenhagen to get more specifics on travel sights. Most of the popular sights are within walking distance of the above hotels, and public transportation is a 2 minute walk away.

You don't need a car in Copenhagen, and I think you are better off without one for a trip of this length. The public transportation is excellent.

I'm not sure I would want to stay right in Nyhavn, but anywhere in the area around it is pretty solid. We stayed at the Hotel Kong Arthur, which is about a 20 minute walk from Nyhavn. We liked it because it was walking distance to most things, and was only a 5 minute walk to the Norreport station, from which you can connect to basically anywhere on the island of Zealand. The price was pretty reasonable too. Alternatively, there are several nice neighborhoods if you wanted to do an AirBnB and save some money. If you do that, I'd recommend trying to find a place near a train/metro station.

As for sites, it depends a lot on your interests. Among our favorites from the city itself: Christiania, Rosenborg Castle (and the surrounding gardens), the National Museum is a must for me, the Ny Carlsberg Glyptotek (an art museum) was great. Carlsberg was just ok, but fun if you love beer. With that amount of time, it might be worth a day trip to Roskilde (cathedral and viking ship museum), Helsingnor (nice old town, Kronborg Castle, and great maritime musem), or Hillerod (Fredirksborg Castle). Check out the Copenhagen Card too - it might save you some money (it includes free public transportation and entrance to most of the places listed above).

For eating out, you have unfortunately picked a fairly expensive city. A few recommendations though. For less expensive meals, check out Copenhagen Street Food on Papiroen (near Christiania) and Torvehallerne (near Rosenborg Castle). For excellent, but not outrageously priced meals, check out Relae (a top 40 restaurant in the world, one michelin star), the Cofoco chain of restaurants (really great prix-fixe menus), and there are some great Neapolitan style pizza places in the city (Mother and Gorm's come to mind). To save money, we ate two meals at Dalle Valle, a very inexpensive all-you-can-eat place (the drink prices are outrageous mind you). The food is fine, nothing special. There are also half price coupons floating around the city, making it an even cheaper meal.

Nyhavn is just one canal lined with cafés, bars and restaurants, it's not really a place with lots of hotels. It's a nice place to visit, especially when the sun is out, but also quite touristy. If you want to have a beer in the sun, do as the local students do and buy your beer at a nearby supermarket or kiosk and drink it by the canal side. If you want too stay in a similar neighbourhood got to Christianshavn, though would guess that it can be a bit expensive.
Steves day trip recommendations are good, check out www.rejseplanen.dk for all public transport schedules and ticket prices.
Copenhagen Street Food and Torvehallerne are also solid advice for cheapish eats. Another place similar to Copenhagen Street Food just opened in the Vesterbro neighbourhood last week, called Westmarket http://www.westmarket.dk/
For non-tourist pubs and bars stay away from the Nyhavn area and most of the city center and go to Nørrebro or Vesterbro.

Thank you all so much! Every piece of information is certainly helpful as I plan out this trip. Keep the ideas and thoughts coming!!!!!!