:w:
It was narrated that Abu Moosa said: The Prophet (peace and blessings of Allaah be upon him) said: “There is no marriage without a guardian.”
(Narrated by al-Tirmidhi, 1101; Abu Dawood, 2085; Ibn Maajah, 1881. Classed as saheeh by Shaykh al-Albaani (may Allaah have mercy on him) in Saheeh al-Tirmidhi, 1/318)
It was narrated that ‘Aa’ishah (may Allaah be pleased with her) said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: Any woman who gets married without the permission of her guardian, her marriage is invalid, her marriage is invalid, her marriage is invalid. But if the marriage is consummated then the mahr is hers because she has allowed him to be intimate with her. If they dispute, then the ruler is the guardian of the one who has no guardian.”
(Narrated by al-Tirmidhi, 1102; Abu Dawood, 2083; Ibn Maajah, 1879. Classed as saheeh by al-Albaani in Irwa’ al-Ghaleel, 1840)
Regarding proposing to a woman with kufaar parents, or a wali that's a kufaar:
http://islamqa.com/en/ref/389/wali