Is it permissible to give charity to a Muslim transgressor or one who abandons prayer, and to a non-Muslim, and what is the ruling on zakat in these cases?

It is permissible to give charity to a Muslim who is openly disobedient (fasiq) or who does not pray, as well as to non-Muslims who are not actively fighting Muslims. However, it is preferable to give it to a devout Muslim. As for obligatory charity (like zakat), it is not valid for a non-Muslim or for one who does not pray.