I want to marry my cousin, but she doesn't really consider herself to be a Muslim and doesn't practice it. She comes from a good muslim family, as do I, and I practice my faith, and I try and educate her. My question is am I OK to marry her? Does it eventually come down to influence and how my children would be brought up?

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If she consider herself christian, then you can not marry her as an obligatory precaution, in which you can refer to another marjaa, like Sheikh Ishaq Fayadh, but I do not recommend that you marry her, as this will effect your kids badly.
However, if she consider herself as infidel, then you can not marry her at all.