Is insistence in supplication that a girl be destined for the supplicant, despite her engagement, considered an inspiration from God and glad tidings that she will be his wife? And is it evidence of God's will to answer?

Allah's inspiration to a servant to make supplication is a sign that He intends to answer it. Persistence in supplication is one of the reasons for its acceptance. However, the answer is not limited to obtaining the exact requested thing; rather, it takes one of three forms: the supplication being answered immediately, or it being stored for the hereafter, or warding off an equivalent evil. One should not supplicate for marriage to an engaged woman who is content with her fiancé. Rather, one must turn away from her and ask Allah for what is good.