1) Fantasy? Sci-fi? Combo?
Fantasy. My other attempts at worldbuilding were rather scifantasy, but this is close to high fantasy. Though magic is studied like science.
2) Earth? AU Earth? Somewhere else entirely?
Something else, earthlike. If you look closer the flora and fauna are a bit different. Later on it becomes very different. On Earth you can usually trust that a meadow before your house is still a meadow the next day. In Anvar's world this isn't true anymore.
4) If your world has "magic", how prevalent is it?
Magic is quite visible. Every village has at least its own hedge-wizard. And there are magic creatures and spirits. You can talk to your hearth spirit who perceives himself as guardian of your family and you can study magic at the local mage guild (if you have the talent). Not many have the talent for active magic, but passive magic is also possible. Many people are Sensitives and can feel magic instead only see its results.
5) Are the laws of physics and nature the same as "normal"?
Yes, just add the magic science.