Is the paternal uncle entitled to take his brother's daughter from her mother's custody if the mother does not agree to marry him, and is the daughter's going out to visit relatives a sufficient justification for that?

Custody after divorce or the husband's death is the mother's right, unless she lacks the conditions required for a custodian. If the daughter reaches an age where corruption is feared for her, and the mother is untrustworthy, then custody falls to her paternal uncle if he is more capable of protecting and preserving her. However, if the uncle's claim is unfounded, and the mother's place is safer for the girl, then he should not take her. Ultimately, the Sharia (definition) courts are the most capable of resolving these disputes and ruling in the best interest of the daughter.