Is the questioner sinful for going along with his mother in slandering relatives and mentioning women's reputations, and for refraining from congregational prayers—except for Friday prayer—because she dislikes it, despite her need for his presence to remember the number of rak'ahs?

If the mother's mind has been affected by dementia to the point that she no longer comprehends what she says, then the pen has been lifted from her, and she is not accountable for her words, actions, or shortcomings in worship. This is based on the Prophet, peace and blessings be upon him, who said: "The pen has been lifted from three: from the sleeper until he awakens, from the child until he matures, and from the insane person until he regains his sanity or recovers." There is no sin upon you for responding to her in a way she does not intend, as long as she is not rational and will not accept anything else. It is good to distract her with entertaining stories. Prayer is not obligatory for her if she has dementia, and congregational prayers and Friday prayers are not obligatory for women at all. It is not permissible for a man to abandon congregational prayer and pray at home under the pretext that his mother does not know the number of rak'ahs unless he prays in front of her, because she is not obligated to pray in the first place. So let her pray in whatever manner she prefers, and you attend congregational and Friday prayers.