Is it permissible for a woman who broke fasting due to a delivery to pay a ransom?

She should make up for the missed fasting days once she becomes ritually pure, but it is impermissible for her to pay a ransom while being able to fast.

Is it permissible for me to name my daughter "Leen"?

“Leen” means “ease”, and there is no harm in giving your daughter that name.

Is it permissible to fish on Saturday?

All Perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions. 

It is permissible for Muslims to fish on any day, but the sons of Israel are the ones forbidden to fish on Saturday. Moreover, there is no evidence in Islamic Sharia prohibiting Muslims from fishing on that day. And Allah Knows Best.

I`m infatuated with a man, is it permissible for me to make supplication that he falls in love with me and becomes my husband?

Islam has honored women by being proposed to by men, and not the other way around, so it is inappropriate for a woman to propose to a man since one who hastens in asking for a thing prematurely shall be punished by deprivation. In fact, such a phenomenon is the result of unlawful mixing between the two sexes.