What is the expiation for breaking fast due to being on a journey, or being sick, or being in a state of menstruation?

No expiation is due on the aforesaid categories, but they are obliged to make up for the missed fast. However, if any of them failed to do so while being able to, and the next Ramadhaan has come, then making up for those days is incumbent on him/her , and paying the ransom as well. And Allah Knows Best.

Are minors obliged to fast?

A male under the age of puberty isn`t required to fast, but his guardian should order him to fast if he was able to endure the hardship until being used for fasting in general. And Allah Knows Best.

I wish to Pray at the Mosque, but I couldn't because I am a Woman. How can I get such a Reward?

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                                                                                                                                                                                                                            Abdullah Ibn Masood (May Allah be Pleased with him) reported that the Prophet (PBUH) said: "Her prayer inside her home is better than its courtyard, and her prayer in her own bedroom is better than inside her home." [Abu Dawood]. Her staying at home, and not going to the mosque in obedience to Allah`s injunctions entails a great reward similar to that bestowed on a man for going to the mosque since the Mercy and Blessings of Allah Prevails. And Allah Knows Best.

A man who was on travel prayed Duhr as four Rakhas upon leaving Tafilah heading to Amman. However, on his way to Amman, he prayed Asir as two Rakhas (Shortened). Is what he did correct from an Islamic perspective?