"Why do I have to pay?"
Translation:Чому мені треба платити?
Yes! ‘Я мушу/маю’ sounds more natural here. And BTW, since ‘платити (to pay)’ works in both directions, ‘чому мені треба платити?’ can be interpreted as ‘why do they have to pay me?’ :) It's not that you will be misunderstood that easily but I'd rather avoid such ambiguity.
P.S. I see now that the topic says ‘Чому я маю платити?’ But I've come here from the ‘Чому мені треба платити?’ so that's how I was answering your question.