1. Is this Story real or Tv Shows Made up?
This story can be found in Matsya Purana too.
2. Dharma behind killing sage Birgu's wife for protecting Asuras?
3.Giving shelter and protecting the guest is one human's duty isn't it and killing him is confusing
As mentioned in this comment, "Supporting Adharma knowingly is also Adharma." Also, sage Bhrigu's wife made Indra completely immobile and was about to burn both of them (Vishnu and Idra) up through her powers. So Lord Vishnu severed the lady's head in self defense.
Here is the summary of the story from Matsya Purana:
Shukracharya consoled the demons. "Do not worry," he said. "I will try and acquire powers that will make the demons invincible. I am going off to pray. While I am gone, do not fight with the gods. Give up arms and lead the lives of hermits. Wait till my return."
Meanwhile, the gods got to know what Shukracharya was up to. They realized that, once Shukracharya returned, they would be in no position to tackle the demons. The best thing to do was to attack the demons immediately, when they had given up arms and were living as hermits. The demons tried to tell the gods that this was not fair. They should not be attacked when they had forsaken arms. But the gods would not listen. They started to kill the demons. The demons fled to Shukracharya's mother, Bhrigu's wife, for protecion.
When the gods attacked, the lady used her powers to make Indra completely immobile. Indra could not move at all. He stood there like a statue. This strange sight unnerved the gods so much that they started to run away. Vishnu came to help Indra. He told Indra to enter his body, so that Vishnu might be able to save him. "I will burn both of you up through my powers," said Shukracharya's mother. "What are you waiting for?" Indra asked Vishnu. "Can"t you see that this woman will destroy us both? Kill her at once." Vishnu summoned up his sudarshana chakra and with this, he neatly severed the lady's head.
The sage Bhrigu was not present at that time. When he returned and found out what had happened, he was greatly angered:
Vishnu had committed the crime of killing a woman. Bhrigu therefore cursed Vishnu. As for his own wife, Bhrigu resurrected her through his powers.