I guess we cannot directly translate English into Japanese and vice-versa. I think that is a common mistake made by a lot of English speakers. You have to think in Japanese instead. Let me use some of your examples and provide some examples:
As a general guide, the thinking process for te-iru forms is "being in the state". For 1, it is very direct. Transitive verbs in the te-iru form can be directly translated into V-ing. Hence, I am opening the door. You can think of it as, "I am in the state of opening the door."
For 2, similarly, think of it as being in the state. So you will get, the door is opened, it is in the opened state. The next question is, how to express "the door is opening (by itself)?" That would be just, 「ドアが開く」. Remember, do not think in English. If you think in English, you will be confused. 「開く」just means open, the act of opening. Hence, naturally「ドアが開く」can be processed as "the door opens". However, it also means "the door is opening"! You should know by now that Japanese depends heavily on context. So, let's put this into context.
The elevator just arrived. ドアが今開く。
The elevator's doors open now/are opening now.
If you really want to express the continuous form, to emphasize on the continual action, then you could also say ドアが開いているところです。
Let's move on for more examples.
For 3, similarly, thinking of the state, the dog is in the state of being dead.
For 4, it is translated as the plane is flying... now why is a "V-ing" used here? It is because flying is the state of being in flight! Hence, it makes perfect sense!
And lastly for 5, "He is laughing"; he is in the state of laughter.
Allow me to emphasize the point of being in context again. For example, 「私が怖い」can mean "I am a scary person" or "I am afraid". 「猫が怖い」can mean "I am scared of cats (私は)" or "Cats are scary". So Japanese language is actually very ambiguous and depend heavily on context. Take some time to ponder over it and you will eventually get it!