american english - How to report a question, both directly and indirectly

So, I'm walking down the street with a friend, who learns that my brother-in-law is a barber, and I go there to get my hair cut. My friend says,

So, do you pay for those haircuts, or does he do it for free?

I think I understand how this works, but I want confirmation. If I describe this conversation using direct speech, I would write:

He asked me, "Do you pay for those haircuts?"

but indirectly, I would write:

She asked me if I paid for my haircuts.

(Assuming I did this correctly, I noticed two things changed: the word do changed to if, and the tense of the verb pay changed from present to past).

Here is my question: Would it be right or wrong if I tried to rewrite it something like this?

She asked me did you pay for them.

That sounds wrong to me, especially without any quotation marks, but I want to make sure I'm understanding how the English changes between direct and indirect speech, particularly when the reported speech was originally a question.