From the Polish guy perspective:
Would they potentially not want me to teach them?
Because you are black? On the contrary - I think that it may give you some extra students. You know - you look like a "real foreigner".
Will my race be an issue in day to day living?
It depends how you define "an issue".
First - you have to understand that the Political Correctness is virtually unknown in Poland. On the West people are learned to be "color blind" (or at least to pretend). Not in Poland. Therefore from time to time you may be refered as "the black one". Exactly the same way you may be called "baldy, red-head or fatso" if that would distinguish you from the others.
Second - average Pole has no prejudice towards blacks. Simply because there was so little contacts with them. There are no black communities in Poland or something like that. I think that most knowledge about blacks in Poland are from the Hollywood films. Therefore you may expect questions like: "Can I see you gun?" or "Can you prepare a voodoo doll for me?"
If you are over-sensitive on these points, it may be a source of real issues.
My advice: Trust your sense of humour.
You are from NY, so you know what to do and where to go and on what time of the day, right?
Thugs are the same all over the world. Your chances of being mugged or beaten are slightly higher than the average. As a black person in the white community you are stand out of the crowd. As a foreigner - you will be seen as a rich guy (worth of mugging). The pepper-spray suggestion is the good one IMHO.