Is it permissible for a paternal half-sister to hug and kiss her brother and to be alone with him, especially after the father's death?

A half-sister through the father is a mahram, whether the father is alive or deceased. Kissing (on other than the mouth) and embracing are permissible between them as long as there is no suspicion of impropriety. However, if there is suspicion of impropriety or fear of temptation (fitna), then kissing and embracing become impermissible. In fact, even looking with lust between them is impermissible by the consensus of jurists.