1) If a Muslim man gets married to a Christian girl without converting her to islam, and this marriage is conducted according to Christian ceremony without the nikah ceremony taking place, would the marriage still be valid?
2) Can a Muslim female get married to a non-Muslim male?
1) Nikah has two essential components: willingness of the bride and bridegroom and an announcement before a group of people that it is taking place. If Mahr (dowry to be given by the bridegroom to the bride) has not been decided, it will be implied that a reasonable Mahr has been decided. It is always advisable to decide about it before marriage. The rest of it has to do with either certain things that make the ceremony more Islamic in spirit (like khutba) or with the customs of the society. However, nothing else is mandatory. In my opinion, therefore, if some formal process was followed to get the marriage formalized even though not exactly the way we normally do as Muslims, the marriage is valid from an Islamic point of view.
The Almighty has allowed Muslim men to get married to Jew and Christian women. The Quran says: “… marriage with chaste free believing women and also chaste women among the People who were given the Book before you is made lawful for you, provided that you give them their dowries and desire chastity, neither committing fornication nor taking them as mistresses.” (5:5)
2) Muslim women cannot get married to non-Muslim men.