The correct answers are She is going to have her dress altered. and She will get her hair styled by a professional.

We use have/get + object + past participle to say that we arranged for someone else to do something for us.

"Having her nails to paint" is incorrect because it uses the infinitive ("to paint") instead of the past participle ("painted").

"Will have altered her dress" is the future perfect tense, which means Sarah will do the altering herself!