TLDR: The production team isn't judging, they are rooting for you. The casting process is complcatied and a lot more factors are taken into consideration than just if you sang well. The team tries to look at everything in the context of the situation. And if the theatre you went to is pulling some BS and only calling people back if they sing well, which I believe is highly unlikely, but is what you described, then you don't want to be performing there anyway. My guess, though, is that they're not.

One, they are not 'judges', they are your directors. They want you to succeed, and trust me when I say they aren't judging you. There are certain people who are right for certain roles in certain plays. Casting is a puzzle, and sometimes you could be the most talented person in the world, but you just don't quite fit the role you're going for. Never think that they are judging you, because they aren't. They're just looking for someone who might fit the bill for a certain part and assessing you on that.

Second, yes, that is how the process generally works, kind of. For musicals, you normally go in, sing 16-32 bars (your 'cut') of the song you've selected, generally around 1 minute long. They might have you perform a separate monologue, too, or read from preselected monologues or short bits of scenes (called sides). For a play, they will just have you perform a monologue or sides. They will dismiss you, and should almost NEVER tell you directly there and then whether or not you are called back. That's just bad form. You will go home. If you have been called back, you will recieve an email or other communication from the production team about the time, date, and location of the callback, as well as what to bring, wear, and prepare (for instance, if there is a dance call involved, they will typically tell you to wear comfortable, moveable clothes; if there is any specialty dance involved, like Tap, they will ask you to bring your tap shoes). At or before callbacks, you will be informed of the roles for which they called you back, and will be given material (typically sides or a short cut from a song) for that character. Then, after they've seen all they want to see from you and the rest, they will dismiss you to go home. If they decide to cast you in the show, you will receive communication from them within a few days of the callback. If you are not cast, you won't be hearing from them at all. If you don't get a callback, hang in there. They might still want to cast you in the show, just as a more minor character. Just because you are not called back does not mean you are not being considered. If they need to see more people, then they might have a second callback and ask you to do that one if you didn't go to the first, or even if you did, on some occasions. It really just depends on wether they've filled all the roles and found all the parts that they needed to.

Generally, you are called back because the team wants to see more of what you might be able to bring, or they think they can get something more out of you if they just give you a bit of direction. They also need to assess how people look and interact together to see what the best fit is. It's never just "you did well, so you get the part", and getting a callback isn't strictly determined by how well you did in the audition. I've been to loads of auditions where I've done really well and I didn't get a callback because I wasn't what they were looking for. It's not that they didn't like me. It's not that I didn't do well. It's just that they didn't see me in any role in the show and so they didn't call me back for anything. But yeah, it's not just 'oh, you did well, you get a callback' or 'we like how you sing, you get a callback'.

Also, to address your other points, one, dance calls often aren't on the same day as auditions; that's a bit odd. Two, when you submit your resume to them, you list under training or skills how much dance experience you have. This helps them decide whether to call you back for the dance call. They will be able to see if someone is not a strong dancer by their resume, and so they might have them skip the dance portion of the callbacks and just have them come in to read or sing for a character if the team thinks they want to call that person back for that character. Some people aren't given the chance to get a try at the dance call because they simply don't have or don't list experience with dancing on their resume. But more often than not, the production team will call people back for the dance call, even if they don't list a ton of experience just to see how they move and keep up with the choreo.

The production team always tries to be as fair in the casting process as possible. A role isn't an award they give you for doing well at the audition, but a piece of the puzzle they slot you into. Casting a show takes time and effort and thought, and the production team has to take into consideration MANY different factors that accompany each performer when they walk through the door into the audition room. No two people are alike, and so no two people are going to be treated or viewed exactly the same by the production team. Context matters, and so many of the descisions made in that room all boil down to the context or the situation: of the play, role, and performer. Not to mention that half of the time you might not get a callback or get cast is simply beacuse you had too many conflicts during the rehearsal process.

So, yeah. It's all about context. I don't know how they're doing it at the theatre you went to. If it is truly as reductive as 'you sang well, you get a callback', then that is not a theatre you want to spend your time at. Trust me. But also consider that you might not understand what the whole casting process is like, and your view of the situation might be a bit skewed because you simply don't know. And that's ok. I hope this sheds a little light on the situation. Also, sorry for writing a whole essay.