I hate to avoid answering your question directly, but (1) I was on a 9-night itinerary, so 3 shows wouldn't have been excessive for me and (2) I only did a couple of the main shows on my cruise. I just wasn't interested in some of them. I went to Rock of Ages and walked out; and I went to Burn the Floor, which I loved; and I went to the magician, which was impressive from a skill standpoint, but not from a show-performance standpoint.
Having said that, I can tell you from my DCL sailings that I loved having 3 shows on a 7-night, 4-port itinerary. It definitely wasn't too much for me, personally. If you like the shows, then it's not too much. For me, personally, I don't think it has to do with the number of ports on the itinerary. (If I'm putting too much emphasis on the "four ports" part of your question, I apologize.)
I was told yes, but only on port days. And you can't get it ahead of time. And they might not offer it in every port.
No.
I'm not 100% sure, but I strongly suspect that you will not.
And I'm sorry, but I have no idea about this one. Do they require parents to attend an Open House normally? I hadn't heard that.
You need a good travel agent!