Is the sister obligated to be kind to her brother and his children if he is unemployed and she refuses to help him, and is his bad treatment of her and their quarrel considered a severance of kinship ties?

One must repent to Allah for severing ties of kinship, for it is one of the major prohibitions, and a severer of ties of kinship will not enter Paradise.

If the enquirer is able to earn a living, then his sister or others are not obliged to support him; rather, he must earn a living to support himself, his wife, and his children. If he is unable to earn a living and has no money, then most scholars hold that his sister is not obliged to support him and his children.

He must maintain ties with his sister and seek her رضا (pleasure/satisfaction), strive to earn a lawful livelihood, and rely on Allah.