It is normal behavior but not acceptable. If it were my child some thing I would do to help break the habit are...
1) Give hugs and stuff when she is not throwing a fit. it sounds like she may need more attention and gentleness when she is transitioning. That is fine and normal and should be indulged. She is clearly having a hard time. Throwing a tantrum to get it is not fine. Give the hugs, attention, affection and be aware of the situation to head off tantrums.
2) not offer pity for screaming. I always told my kids, quite calmly to "stop screaming" and carried on with what we were doing.
From the examples you gave.
With the car door I would have told her to stop screaming and then asked her if she wanted to get in the car herself or would like me to put her in.
With the stool incident I would apologized for what the baby did and offer her some calm words. I would have given a hug and reassured her that everything was ok now and told her to stop screaming. Once she calmed down I would remind her how to ask me to move the baby.
With the toy I would have probably let her keep it if the baby was not upset. Depends on how often this happens etc. I would have reminded her not to take toys away from the baby and told her what the consequences will be if she does it again (The baby does not seem to mind right now but if you do it again we will have to give the toy back to the baby. Please don't take toys from the baby, If you want something she has ask me first and I will help you.) I may have asked her to get a new toy for the baby before she went on playing. Once she was tantruming though I would have probably put her in time out (maybe on the couch in the other room) but it sounds like the mom is not into that so I would have just let her scream and told her she needs to stop screaming and when she was done she may come and have fun with us. I mean really a tantrum is its own time out. She is effectively taking herself out of the fun.
Oh, and I agree with the poster above me....Just because I have words doesn't mean I can always use them. I am prone to have a little hissy fit every now and then and I am 36. When people get angry, frustrated, overwhelmed it is hard to calmly say what they are feeling. Sometimes we just snap. That goes for very verbal 2 year olds as well as adults. Tantrums happen and it is a life long process to learn to keep them from bursting out. And even when I use words they do not always come out in this serene and calm and pleasant. Sometimes they come out in a very well worded tantrum

Still completely unacceptable but we just have to learn to control our outburst. That doesn't mean they go away.