What is the ruling on seeking European citizenship for children to facilitate administrative matters and safeguard rights, knowing that the father obtained it out of necessity, and his children still hold the citizenship of their original country?

The basic ruling is that it is forbidden for a Muslim to voluntarily acquire the nationality of a disbelieving country, unless there is a necessity or an overriding legitimate interest. It is not possible to determine whether necessity or interest applies to your children, as your children acquiring the nationalities of disbelieving countries and growing up in them may lead to their deviation and apostasy from the religion.