What advice can you share for a short February trip to Norway?

As a local who has lived in Tromsø before... It's kind of hard to appreciate what you have (I didn't go do any of the tourist things when I lived there, I was busy studying at the university). However, I can give you a couple of tips:

• As for aurora hunting, you don't need a guide. They can't control the skies. All you need is favorable atmospheric conditions, a relatively cloudless night, and an area with little light pollution. I happened to live on Kvaløya, an island off the main Troms island which is like a suburban area. It's a quick 15 minute bus ride from the city center. This area is excellent for watching the northern lights, as the island is basically a mountain with a few houses along the bottom. You can easily get off on the bus stop just over the bridge (I know this because I lived in a house like two minutes away from this bus stop) and just walk up into the mountainside, which should shield you from light pollution and give an excellent view of the aurora if it's out. Google aurora forecast to find the best date. Also unlikely to meet any other tourists.
• There's also a cable car line called Fjellheisen that goes onto a mountain on the mainland side (Tromsdalen). I never went but it's supposed to be a spectacular view and also a good spot to watch the aurora, albeit more touristy.
• The Arctic Cathedral (Ishavskatedralen) is also on the mainland side, and probably the most iconic landmark in the city. It's located right near a bridge so it's possible to walk there from the city center.
• Other attractions include Polaria (an aquarium that houses seals among other things) and the Polar Museum, which exhibits Norway's history of polar exploration as well as exhibits dedicated to historical whaling.
• If you're looking for restaurants, a local favorite is Skarven, which serves seafood and steak. It's a bit on the expensive side (not Michelin star expensive, but not cheap). Or if you want to cross an item off your bucket list, Tromsø also has the world's northernmost Burger King (whose menu is exactly what you'd expect).
• If you want to sample the local youth/student scene (I don't know how old you are or if you'd feel too awkward), the local student union has a venue called Driv. It frequently hosts concerts and also has a bar. Although it is a student focused venue it is open to anyone.
• If you get the chance, definitely sample some reindeer meat. It's unique to the northern part of the country and can be quite delicious if prepared well.