Is it way as intended or is it a bug that a duke who does not have an estate is unable to join any of the realm's armies? I simply can't find the option for it any longer anywhere in the game menu.
This is a known issue. It will be addressed eventually. The exact mechanism for Dukes/Rulers to join armies is still not yet determined.
Oh, something that should maybe be mentioned about the situation as well: For some reason, one of the lords of my duchy is shown as my character's liege in the "Hierarchy" part of the list of who I can send messages to. Is this a bug?
Yes.
