Is the father's consent required for marriage, and is he the sole guardian? And what should the daughter do if he insists on refusing, especially with his separation from her mother and his actions that are offensive to religion?

For a marriage to be valid, the woman's guardian must give her in marriage, based on the Prophet’s (peace be upon him) saying: “There is no marriage without a guardian.” The most rightful guardians are the father, then the paternal grandfather, then the son, then the full brother, then the half-brother (from the father’s side), then their sons, then the paternal uncles. The maternal uncle is not among the guardians. The guardian must be Muslim, adult, sane, mature, male, and free.

If your father is considered an infidel due to his complete abandonment of prayer, his guardianship lapses and transfers to the next in line. However, if he prays sometimes and abandons it sometimes, he is not considered an infidel, but rather a transgressor (fasiq). The preponderant opinion is that a transgressor may be a guardian. Your father must give you in marriage, and your marriage contract is not valid without his approval, unless he is preventing you from marrying (ʿāḍil), in which case the matter should be referred to the judiciary.

We advise you to perform Istikhara (prayer for guidance), be open with your father about the suitor, try to convince him, and ask Allah to relieve your worries. We also advise you to fear Allah regarding your father and treat him with kindness, as Allah Almighty says: "And to parents do good" [Al-Isra: 23], and the Prophet (peace be upon him) said: "The Lord's pleasure is in the parent's pleasure, and the Lord's displeasure is in the parent's displeasure," and "The father is the middle gate of Paradise; so, if you wish, abandon that gate or preserve it."