Is it safe to use a flat iron to get the stray lice eggs out of my daughter's hair?

Wouldn't advise it. If you use vinegar (regular, apple cider vinegar or pickling vinegar)as a hair rinse it will soften the glue that holds the nit to the hair & it will separate easier from the shaft.A fine tooth nit come is helpful for this process. It can be repeated daily for a week & may kill active bugs too.
No! That sounds dangerous to me! there are special very fine-toothed combs meant to remove nits (the remaining lice eggs) from strands of hair.