What is the ruling on giving one's mother a car or helping family members build a house, if it is used in ways that violate the conditions of the donor, namely that it be used for obedience to God, and would the giver be sinful or an accomplice to the transgressions that might occur on their part?

Giving a car to your mother and siblings for their permissible needs, and enjoining them not to use it except for what is lawful according to Sharia, is a good deed. You bear no sin if they use it for something other than that. Likewise, assisting them in building a house; you will be rewarded for it, and you bear no sin if they do something unlawful within it.