1. I imported a few files for my game and got the model to work - it's untextured for now. Not sure how to get the texture to be applied in the engine though. But anyway, the problem is that the shadow appears oddly on this building model. I'd love to test if the problem happens with other models, but the other models aren't necessarily complete and I don't want to expose too much to the public. Anyway, here it is:
I turned on Diagonal Light so the shadow can be more dynamic and easier to see the problem. As you can see, the shadow is appearing a little on the well lit side of the building even though it shouldn't be there since the other side is totally darkened. What's the problem with it?

2. Shadows do not appear dynamically on the terrain from objects placed in the World Editor. Why is this?

3. Shadows in your engine are animated, right? Is there an option to allow players to choose inanimate shadows (little circular shadows under character and static shadows from objects) to support low-end computers better?

1. this is not about shadowing but about shading/lighting the model, more exactly you've got bad vertex normals
when exporting from your 3d software to (.obj or .ase or .x or .xml) make sure you select "export vertex normals"

2. World Editor has shadowing disabled, you will see full shadowing when you'll run the world in your own code

3. yes all shadows are dynamic, I'll be working on adding static shadows (in form of lightmaps) soon