Why is no instrumental case (dobrou studentkou) used here?
Because it is not necessary. Both the nominative and the instrumental are equally possible here. There are some rules for preferring one or the other but they are an advanced topic and they are not strict.
Děkují mnohokrát! Jedna z mých nejdůležitějších vět je „jsem cizincem“.
In that one I would actually prefer the nominative "cizinec".
Yep I found the instrumental case expressing a bit imperfective meaning. I.e. : I enjoy beeing a mother. Baví mě být matkou. (Nominative wouldn't work well here)