Is telling a friend about some of her brother's sins considered backbiting? And is it necessary to make up a day of Ramadan if one experienced a wet dream during sleep and was ignorant of the ruling on ghusl?

Your friend's discussion about your brother's sins is backbiting unless he openly committed them, and backbiting is forbidden in Islamic law. She should have concealed your brother's sins, and you should ask her to stop talking about them. Nocturnal emission (ihtilam) in Ramadan does not invalidate the fast, by scholarly consensus, nor does it break the fast; because the Prophet (peace be upon him) said: "Three things do not break the fast of the fasting person: cupping, vomiting, and nocturnal emission." As for prayer without purity (tahara), it is invalid, whether it is major purity (ghusl) or minor purity (wudu). Major ritual impurity is not removed by wudu, but by ghusl. The prayers performed without purity are not valid, and it is hoped that you will not bear sin for being unaware of the ruling.