Is it permissible for a father to use a company car for personal errands and to fill its tank at the company's expense, and is money earned from a business venture, whose account was opened using this car, considered ill-gotten gain?

There is no objection to using a work vehicle for private purposes if the employers permit it and cover its expenses, and this permission includes the children. However, if they do not permit it, then it is not permissible to use it for private purposes, because that is an aggression and consumption of wealth unjustly. The decisive factor is the work system and its laws, for "Muslims are bound by their conditions." Using the vehicle unjustly does not affect the permissible transaction that you engaged in, but you must repent and restore the right to its owners.