Is it permissible to live with a Christian mother who intends to marry a Christian man, knowing that the questioner is a Muslim and her Muslim father has passed away?

It is permissible for you to reside with your mother and her non-Muslim husband, for he is a mahram to you, and you are not obliged to veil from him. It is also permissible for you to travel with him according to the majority of scholars, as long as he is deemed trustworthy, though the Hanbalis do not consider him a mahram specifically for travel. A mahram is one whom it is permanently unlawful to marry due to kinship, suckling, or affinity, and this includes both Muslims and dhimmis. Our advice to you is to strive in inviting your mother to Islam, for in it there is great reward, and it is a form of righteousness and kindness towards her.