I heard that if I apply for a residency visa, I automatically have permission to work. Is this correct?
You only have permission once the residence permit is granted - not before.
Additionally I would appreciate any information on job leads in Rzeszow, the demand for private English lessons and the going rates for an experienced teacher.
Job leads? Forget it - Poland works very much on being seen in-person. The demand? As in any other European country, except those with a great command of English. Going rates? Incredibly variable - however, I have my doubts that you'll be treated as anything other than a first-year teacher. [
Which is an easier niche in Poland, teaching English for business or for children? Specifically in Rzeszow? What is the salary difference? Can I teach English to children if my Polish is basic conversational with grammatical errors?
Neither is easier - both are demanding. Salary difference? Impossible to say - I was once dealing with an offer of 100PLN/hour to teach a 2 year old child. Can you teach children? Only you know.