Another word for no regrets is impenitent.
Impenitent means to not have any regrets or feel any shame about one's attitude or actions.
Examples of impenitent in a sentence :
-The bully was impenitent when he pushed the little boy off the swings.
-Julie's impenitent attitude after reading Sara's diary in front of her whole class, led to detention from the Headmistress.
The sentence that had correct sentence structure is
Before we were able to get financing we had to have the house appraised by independent appraiser