Is a girl considered a virgin because her hymen has not been broken, even though intercourse has taken place, and is it permissible to take her back verbally or with a new contract?

You erred by resorting to a female doctor to ascertain your wife's virginity and by having ill thoughts about her. This does not permit the exposure of private parts. You must repent and seek forgiveness.

It is not a condition for blood to be shed upon the breaking of the hymen. Your divorce through the registrar is effective, and the statement that she was a virgin is incorrect. You can take her back before the expiration of her waiting period (iddah) if you have not divorced her three times. The waiting period ends upon purification from the third menstrual cycle or after three months if she does not menstruate. The return (raj'ah) is effected by word or deed.