I'm not a native Greek speaker, but I do know the answers to your questions.
1. Με πονάει το στομάχι μου. => My stomach hurts ME. "Με" is the direct object (accusative case) of the verb "πονάει".
2. Word order in Greek can be flexible. Putting "με πονάει" first emphasizes the fact that this person's stomach hurts. The subject (nominative case) of the sentence is "το στομάχι".