She was an only child and indulged at every whim by her loving father.
Is 'the" more appropriate before only child?
No. There is a strong collocation: 'indulged <possessive-personal-pronoun> every whim'. This archaic word-order is preserved in expressions such as 'She was an only child, and her doting father indulged her every whim.'
Your 'loving' would work too, but 'doting' seems more appropriate to me in this context.
b
There is no passive voice in your sentence. You can embed it in here by saying:
She was an only child, and her every whim was indulged by her doting father.
But I wouldn't do that.
BN was right - there's no passive here. But I think what you may have in mind is that in my example I changed the object of 'indulge' from the girl to the whim. You can indulge either a person or that person's wishes; the verb 'indulge is flexible like that. But the collocation I suggested isn't flexible at all. That's why I made the change you may have noticed.
b