Please ask your pediatrician's office today for some recommendations for good books on toddler developmental stages. This should help reassure you that this behavior is totally normal. He's not "over emotional" -- he is two years old, barely out of babyhood and still not able to understand much beyond "I want what I want when I want it, and I want it NOW." Children this age are often easily frustrated (I want to open the door and I can't), easily jealous (You have MY toy and I want it back even if I was totally ignoring it until I saw you with it) and so on. He does not have any experience to tell him that there are ways to react other than by getting upset.
Most of all -- please do not expect more from him than he is able to achieve at his age and his stage. If you think a two-year-old should be able to control his emotions when another child takes a toy, please do read some reputable books about child development and you'll see that he is just not there yet and no child his age would be there yet. Through well-informed parenting you'll teach him how to channel frustrations and cope when things don't go his way. For starters, distract and redirect him instantly in the situations you listed here: If a friend's playing with his toy and he suddenly wants that toy, distract his attention with another one. And so on.
It's not a matter of needing sleep or being "bad" or wanting intentionally to upset mom - it is typical for a child of two who is still learning about his world.