Is the mother's decision to travel for Hajj solely with her father (the questioner's grandfather), so he can teach her the rituals, a sound one, despite the availability of independent travel from Europe? And what is the way to perform Hajj when financially able without relying on the lottery?

It is not permissible for a woman who believes in Allah and the Last Day to travel except with a mahram (definition). Therefore, the mother's decision to prevent her from traveling to the lands of the disbelievers and then performing Hajj without a mahram is correct, based on the Prophet's (peace be upon him) saying: "It is not permissible for a woman who believes in Allah and the Last Day to travel except with a mahram." If she is able to travel with her father or another mahram without resorting to a lottery, then that becomes obligatory for her. The son should hasten to repent to Allah Almighty, for: "And the record [of deeds] will be placed [before them], and you will see the criminals fearful of what is therein, and they will say, 'Oh, woe to us! What is this book that leaves nothing small or great except that it has enumerated it?' And they will find what they did present [before them]. And your Lord wrongs no one."