Are the pieces of advice given to a wife—to pray for her husband, to remind him of Allah, and to take care of her appearance—considered correct in her dealings with a husband who is cheating on her and threatening her with divorce?

What she was advised is correct, and she should not despair of advising him to take all legitimate means to convince him to give up the forbidden. If he insists, then there is no good for her in staying with him due to his wickedness and bad companionship. And he should know that what he is doing are major sins, and Allah has enriched him with his wife. Provision is guaranteed, and what he took from that relationship is forbidden, so let him fear Allah and repent to Him.