Is what the questioner did, getting to know his female cousin online with the intention of marriage, permissible, taking into account the Islamic legal guidelines, and is it permissible for him to continue corresponding with her until he proposes to her?

Correspondence and conversation with a non-mahram woman are not permissible. The religiously prescribed method for engagement is the best. Correspondence does not reveal true character; rather, it incites lust and instills a love for zina (unlawful sexual intercourse). Therefore, it is advisable to protect oneself, one's religion, and one's honor by refraining from it. The suitor is only permitted to look at the woman he intends to marry. It is permissible to عقد القران (perform the marriage contract) and delay الدخول (consummation of the marriage) so that meetings may be lawful and acquaintance deeper.