Let's see... I think you may have mentioned on X that you were stuck when you get back to Robin's house? To move to the next step there's just one crate in the lower right side of his room that you need to "check" to progress. I probably need to make that a bit more obvious in the next update.
It's quite possible that you have run into a bug somewhere. Would be great if you could send me a screenshot of some of these issues either on X messenger, email (zimogu@proton.me), or just post it here and I'll patch it asap.
As for some of the other issues, overall, you play as the princess for a very brief period of time and you have a very specific task to do [don't want to spoil it in the comments here]. Re: the room on the left, that's the princess' parents room and she's very opposed to going in there uninvited! All you need to do when playing as the princess is walk upstairs and game should progress.
As for the invisible king and queen, I'm pretty sure that they've been set to "disappear" when you come back downstairs on the way to the basement. Actually they are technically "away" for a brief period of time and should be seen again later in another scene (though you are escaping from the castle at that point and have no need to talk to them).
Hope some of that helps. Feel free to reach out.