How should a woman make up prayers that she missed two days from delivery?

All perfect praise be to Allah,The Lord of The Worlds                                                                                                                                                                        It is prohibited to delay prayer beyond its prescribed fixed time, therefore, the woman in question is obliged to make up missed prayers after delivery and confinement i.e. once she becomes ritually pure. And Allah Knows Best.

Is it permissible for the person who wants to travel from the United Arab Emirates to Jordan during the daytime in Ramadan to break his fast before leaving his country; knowing that he had embarked on journey half an hour after dawn?

Praise be to Allah the Lord of the Worlds.

The person who made the intention to embark on journey after dawn must observe fast for that day, because it became an obligation on him before engaging on journey. However, if he experienced unbearable hardship while on travel, then he is permitted to break his fast, but he must make up for the missed day. And Allah The Almighty Knows Best.

Is it permissible for someone to make a supplication to Allah to marry an already married, or an engaged woman?

No, it isn`t, and the person who does so is sinful before Allah, The Almighty.

Which hour coincides with the last third of the night?

This hour could be figured out by calculating the number of night hours which start at sunset, and end at dawn break, then the total is divided on three to get the result.