She would fit in very well. The Fairy Tail guild is all about family and acceptance. Given her parents are Lucy and Natsu, she'd be welcomed with open arms. She could learn from the older members like Makarov and Mirajane, and bond with the younger ones like Romeo. Her magic skills would also contribute to the guild's strength during battles or when helping the town.
Lucy and Natsu's daughter would be an integral part of the Fairy Tail guild. She'd grow up in an environment full of love and support. Her celestial and dragon slayer magic would make her a valuable asset. She could form a team with other young wizards in the guild and go on missions together. Her presence would also keep the spirit of her parents alive in the guild, and she'd be a symbol of the continuation of the Fairy Tail legacy.
The guild would likely be ecstatic. Gray would probably make some snarky comment about Natsu finally growing up, but deep down he'd be happy for his friend. Erza would be all for it and might start planning the wedding immediately, giving orders to everyone in the guild to help out.
The guild as a whole would likely mobilize. Erza would start formulating a strategic plan. She'd consider the strengths and weaknesses of Acnologia and figure out the best way to approach the situation. Gray would also be ready to fight, using his ice - make magic to support the rescue mission. They would all be united in their determination to get Lucy back safely.
Sakura could fit in as a healer. In Fairy Tail, having a powerful healer is always valuable. Her medical ninjutsu skills would be highly regarded, and she could quickly become an important member of the guild when it comes to treating injuries after battles.
He could leave because of a misunderstanding. Maybe he overhears something wrong and thinks the guild doesn't trust him or value him anymore. So in a fit of anger and pride, he decides to leave.
Well, it could be really exciting. Their daughter might inherit Lucy's celestial magic and Natsu's dragon slayer powers. Maybe she'd be on adventures with the new generation of Fairy Tail wizards, facing powerful enemies and making new friends just like her parents did.
Their wedding would be a grand affair. All the Fairy Tail members would be there. Natsu would be in his best suit, looking a bit uncomfortable but happy. Lucy would be in a beautiful white dress, with her Celestial Spirits perhaps helping with the decorations. There would be lots of food, music, and dancing. Gray might accidentally freeze something during the celebration, and Erza would keep everyone in line. It would be a day filled with laughter and joy.
He could kidnap her because he believes she's in immediate danger. Natsu is very impulsive and might not have time to explain things properly, so he just takes her away to keep her safe.
Her relationship with Natsu would likely change. Natsu might be initially distrustful of Lucy if he senses her connection to End, since End is often seen as an enemy. But then as he realizes that Lucy is still the same person deep down, their bond could become even stronger as he helps her control this new aspect of herself.
Some of the more mischievous guild members like Gray might tease them a lot. He'd probably make jokes about it every chance he gets, much to Natsu and Lucy's annoyance. Happy, on the other hand, might be really confused at first but then start to see it as a sign of their closeness and be really happy for them.
One possibility is that their daughter has a special ability to combine Natsu's fire magic and Lucy's summoning magic in unexpected ways. For example, she could summon a fire - based celestial spirit. She might grow up in a Fairy Tail that has changed a bit since her parents' time, with new members and new challenges. She could be mentored by some of the older guild members like Erza or Gray. And as she comes of age, she might discover secrets about her family's past that put her in a position to make important decisions for the future of the guild and the magical world at large.