What does (Whoever fasts Ramadan with faith and seeking reward) mean?

Is the marriage, which lacks a valid legal contract, a guardian, and a court registration, valid?

It is incumbent that a valid marriage contract be concluded in the presence of a guardian and two trustful witnesses, and it should be registered in the court to protect the rights of the wife. Actually, a valid marriage contract is what differentiates between sound marriage and fornication.

What are the benefits of slaughtering an Aqeeqah?

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.                                                                                                                                                                                                                          Extending thanks to Allah for His grace, expressing happiness for having a newborn, declaring lineage, and feeding the mother to compensate for the blood that she had lost during delivery.

Can I start fasting based on the moon sighting of another country?

You must follow the moon sighting of the country you are in.