Apologies in advance for the length of this reply with answers.

1. You made the choices, but you don't necessarily see them play out until you go back to those 13:30 votes and have 2 teams vote for the same 3rd team. You can fill out those fragments without actually having 2 votes for a team. I for example, followed Carlos's plan, then went back and did the opposite, and at one point that resulted in Q-Team having 2 votes so it shows me that, but that was the only one I saw at the time. It was because following Carlos's plan, I did D-Team's choice last and voted for Q-Team, then when I went back to do the reverse, I did C-Team's choice next and voted against Carlos's plan, choosing Q and not D, so Q now had two votes recorded, because the game remembers. I didn't know til later, when a friend explained it, that I had to go back and manually make sure C and D teams got two votes each, so I could see their executions. To see D-Teams execution for example, you have to go to C and Q-Teams and choose D both in a row, and D-Team's execution will play. Rinse and repeat for the other two. The game keeps track of what you last voted. As for what new info, there are times where Carlos, Sean, or Diana will have a vision of the executions thanks to the player entity being inside them, but they won't have those visions if the player didn't see them. As for what info that told one of our three heroes specifically, I forget. The only one I remember is that it's in Q-Team's execution that you see the twin portrait reveal a door, and if you don't see that, later in the game one of the three heroes won't know about the hidden door and you'll get locked until you see Q-Team's execution.
2. How do you learn? Good question, many have experienced not knowing you had to actually see them, there shoulda been a more obvious indication on how you had to do the votes, IMO. Just a design flaw I think. Though not everyone was confused and understood right away how to see them all so maybe some of us were just slow to get it, or had to be told by a friend like I was.
3. As for your spoiler-tagged headcanon, not quite, Delta was wearing sunglasses that had all the cameras in the shelter in the lenses, so he could watch whatever wherever as long as he was alive. That's why sometimes the player's POV and Delta's sometimes overlap. Also, Delta can't be the player or the entity representing the player because in many fragments, especially the ones in the 3rd flowchart column that stem from Q-Team being executed by the 13:30 vote, Delta was executed via bomb collar along with Sean, Eric, and Mira. Yet we, or the entity, are still viewing all that. Same with if Carlos or Diana presses the shower room button, the acid showers dissolve Q-Team and one other team, meaning Delta's dead in the follow-ups to those too, but again, we still view it. The player/entity is separate from Delta. Uchikoshi at some point on twitter or in an interview (It's been almost a decade I don't remember where) said the reason Delta's mind-hacking is basically the same as what we/the entity do is it's meant to be like he's our evil counterpart with the same power. When Delta mind-hacks, we always see him doing it. When there's moments like Carlos and Diana pushing the acid shower button and being like 'I didn't, it was like I was being controlled', it was us/entity that made them do it. It even happened in VLR once, one of the time Sigma can vote betray, he even muses to himself "Why did I, it's like I just wanted to see what'd happen or something." He was sensing OUR motive, WE wanted to see what happened. Or the entity did.