@Cantfeelmuchthesedays
Why did you feel you had to say, "I'll just put your dinner down here", when you were going to help DD with the milk container?
It comes across as the subservient comment a nervous kitchen maid would make to the Cook.
You can put a plate down and make your way to the fridge at your own speed without an explanation, and especially without using the word "just".
When DD starts getting worked up about stickers or some other activity, place a hand gently on her shoulder and ask her about her feelings. Ask her if she would like to get up and do a little shaking off of the frustration.
Taylor Swift's "Shake it Off" is a good song to play for a shaking off session, as is "Shake Your Sillies Out" by The Wiggles.
If she doesn't know the word frustration, you need to teach her. If she won't respond, then tell her you are ready to talk about her feelings whenever she is, get up and start doing something else. Say that gently, not as if you've taken the hump. You need to convey to her that you have confidence in her ability to express her feelings and that you're eager to hear what she's feeling.
Sitting with her while and trying to intervene more than once while she gets more and more frazzled is not what a preschool teacher would do. I suspect she is comparing the different leadership styles she sees at home and at school, and that you are (1) on the receiving end of all her confusion and anxiety about school, and (2) comparing badly with a teacher who is able to shrug off her moods or address them more authoritatively.
Think of yourself as a leader in your home, and remind yourself that your child needs the security of knowing that someone older and wiser is in charge.
Keep a leadership diary where you note your own feelings every day. You can share this with your H, and invite him to do a diary of his own too. The two of you can start to develop a leadership style and work together.