How can we tell whether a woman is ‘truly a believing Christian or Jew’ ?

 

Walaikum assalam wa rahmatullah,

There are 3 conditions for validity. She has to belief (without ‘doubts’):

a) God;

b) A recognized revealed Book;

c) A recognized Prophet. [Durr al-Mukhtar Sharh Tanwir al-Absar]

If any of these is not met, the marriage would be invalid. It is invalid (and akin to living in fornication) to marry an atheist or agnostic, even if they are ‘interested in Islam.’

The general ruling of marrying a non-Muslim woman in the West is that it is prohibitively disliked (makruh tahriman) and sinful, as it is not normally possible to be genuinely confident in such a marriage that one's children will grow up as practicing Muslims. If one has this confidence, or is in a Muslim country with solid extended family support, then it is somewhat disliked (makruh tanzihan).
[Ibn Abidin, Radd al-Muhtar]

And Allah alone gives success.

Wassalam,
Faraz Rabbani