Hunting for aurora near Tromso - few questions

The Tromsø region does not lack woodlands. If you go out into Bardu into spectacular Dividalen National Park, there are plenty of woodlands. However, mountain birch forests in winter are nothing like dense forests in Poland or other midlatitude regions. The trees are 3–7 metre high and lots of light passes through, so in most forests your opportunities for aurora viewing are fine. Both in Dividalen and on Senja there are free and open cabins that are deep in the woods (the one in Dividalen will be at least two days walking/skiing, though, and can be tricky to find; I've read some reports of people concluding it didn't exist!).

Yes, you can light a fire. Open fires are prohibited below the treeline between 15 April and 15 September. In February you're fine. Bring your own firewood or collect dead wood from the ground. Do not cut off branches from living trees. In February it may be difficult to tell dead from living trees, so do not cut off any branches at all.

I've hitchhiked in Norway and I've found it works quite well, but I would recommend to try buses as well. The road closest to some of those cabins may be a rural road with little traffic, so it may be a couple of hours before you get picked up. You might be better off roadwalking from the nearest bus stop.

There is no reservation system. For people arriving the same day, it's first come first served. If you've already stayed one or more nights, the group who has stayed the largest number of nights should leave to make place for the newly arrived group. This is even more true if you can sleep in a tent.