Are the actions mentioned in the question—taking one's younger brother into the bathroom naked, touching and embracing him, masturbating afterward, and informing the mother of this—considered apostasy (leaving the faith), or do they require expiation, making up for missed obligations, or seeking absolution from a human right? And is my informing my friends of my repentance considered publicizing a sin?

Congratulations on your repentance. Your act combines two forbidden things: the preliminaries of sodomy and masturbation. Sodomy is forbidden by consensus (ijma'), and its preliminaries are likewise forbidden. These preliminaries are of various degrees, the lowest being looking with lust, and the highest being committing the major indecency. Your conversation with your mother and friends is not considered publicizing the sin, because mentioning a sin to someone whose advice is sought is not forbidden, though concealment is better. This act does not lead to apostasy, and no expiation (kaffarah) is required for it unless you are certain it occurred during daylight hours of Ramadan. If you are not certain that the act occurred during daylight hours of Ramadan, then you are not required to make up (qada') the fast, because the default is that you were not engaged in an act of worship. This act involves the rights of human beings, and the repentant person must seek forgiveness from the one with whom the indecency was committed, in a general rather than explicit way, to avoid stirring up evil.