Is a Muslim is Permitted to Pray Wearing Shoes?

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                                                                                                                                                                                                                                There is no harm for a Muslim to pray wearing shoes or slippers so long as they are pure, since a man asked Anas Ibn Malik: "Did the Prophet (PBUH) use to offer the prayers with his shoes on?" He said: "Yes." Moreover, Ibn Rajab (May Allah have mercy on his soul) said: "Performing prayer wearing slippers/shoes is permissible according the majority of the Muslim scholars." [Fatih Al-Bary,vol.2/pp.274]. And Allah Knows Best.

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 Zakah (obligatory charity) due on the Zakah money received by a poor person, and reached a Nissab (minimum amount liable for Zakah), and a whole lunar year had lapsed over having it in his possession?

Yes, the poor who possessed a Nissab for a whole lunar year is obliged to pay the Zakah due on that money even if it was given to him as a Zakah money in the first place. And Allah Knows Best.

Is buying and storing gold bars for profit permissible or prohibited?