So much about this I don't understand. To start with. Je me suis fait faire les ongles.

Question

Why is it fait not faire or even fais? Also why les ongles and not mes ongles?

Answer

Bonjour Susan,

The basic construction is faire + infinitive = to have something done

Se faire + infinitive = to have someone done to/for oneself

Je me suis fait faire is the past tense = I had something done to myself (fait is the past participle)

Since it's obvious that the ongles belong to me, the definite article is used - see Using le, la, les with body parts

Answer

Thanks Laura,  a big help.

Answer

As to the first part of your question, it is my understanding that someone else is doing my nails (I don't know if I'm right), that's why "fait"--3rd person singular. 
As to "les ongles," just like in Spanish, one does not say "mes ongles" when referring ot one's body part. One says "les."

Answer

Hi Laura, But why is "Elle se fait rembourser. She gets refunded" reflexive, please?
How is getting refunded different from having your house painted? How are you doing something for yourself if you are getting refunded. Thank you! (I find your web site invaluable.)

Answer

Bonjour Mark,
In "she gets refunded" she is the object that the verb is acting upon. She requested a refund, and got it. In "she gets her house painted," she requested that someone paint her house: the house is what's being acted upon, not her.