John Rosemond is a bully, and you sound like you want to be more supportive and caring towards your daughter than that. Of course the puppy deaths have been difficult, even though she might not fully understand. I would continue to encourage and respond to her questions, but also be careful you aren't asking too many questions. Her screaming may just be a way to ask you to be quiet. What sorts of questions are you asking her when she screams? She may just need to be reminded that she can ask you to "please be quiet", if she doesn't want to talk about something. If she continues to scream, I would remain calm, explain that screaming makes it hard to talk, and find something else to do until she is done. Your husband might make a quick suggestion of a better way to greet him, how about a big hug? But if she screams, also matter of factly, "I'll see you when you're finished", and move on. Minimize the attention from screaming, and help her find a better way to get your attention.
As far as the school, I'm not sure why a program for 4 year olds is requiring more sitting and less play. Play is how young children learn, so I would look closely at the school's expectations. What is she supposed to be doing while she is sitting? What does she say was the reason for her biting? Again, help her find a better plan to get what she wants, because biting hurts and gets people angry. Be calm and helpful to her, and if it seems like she is really stressed by a school situation that isn't meeting her needs, find a better school. I also wonder about a preschool that gives a "red" day for a 4 year old child two weeks into the school year, not helpful.