"Do I have my apple?"
"ai-je" is an example of inversion- its a way to ask a question. est-ce que is a more formal way to ask a question. In general, there are three ways to ask a question, in order of formality: 1. Inflection- don't change anything in the sentence, just make it SOUND like a question. (you have an apple?) 2. Inversion- switch the subject and the verb (you have an apple ... have you an apple? it makes sense). 3. Est-ce que- translates as "do" (I have an apple ... DO I have an apple?)
All these types are used in French. Hope this helps!
"Inversion of the verb with the first person je to form a yes/no question is characteristic of ONLY THE MOST FORMAL FRENCH. Many speakers and writers these days would avoid it and use est-ce que. Furthermore, there are idiosyncratic restrictions on its use." French Grammar and Usage, 3rd edition, 2010
It actually has nothing to do with the gender of the speaker; it's "ma pomme" because the word "pomme" is feminine. Every noun in french is either masculine or feminine (ex. LA pomme vs LE livre) and it doesn't matter whether it is a man or a woman speaking, the word will always keep it's gender. So because "pomme" is feminine it will never be "mon pomme" it will always be "ma pomme".