If an orthodox gets married to an unorthodox in court only as in the case of my friend, how does the church regard this marriage? What if rumors have it that one cheats on the other? Is it considered good to inform the cheated end?
Our Church sees marriage between non-church members as civil marriage, though our Church does not recognize the marriage of an Orthodox to a non-Orthodox. In your friend's case, I recommend to contact the Clerical Council in order to decide what she should do since she was not Orthodox when she got married. Let us pray that by her Orthodox faith she would win him to the Lord as St. Peter says "Wives, likewise, be submissive to your own husbands, that even if some do not obey the word, they, without a word, may be won by the conduct of their wives, when they observe your chaste conduct accompanied by fear" (1 Pet 3:1,2).
Your question about rumors is not clear enough.
St. James teaches us "If anyone among you thinks he is religious, and does not bridle his tongue but deceives his own heart, this one's religion is useless" (James 1:26). The best thing is to pray for her and her husband that God be with them and lead them to live the life of purity and righteousness.