Aslan2's explanation is exactly the one I had at school when we were taught about the Şart kipi. I'd just like to add that İ see it this way:

Bana bak! - EMİR
Baksana bana! - TATLI EMİR.

Usually you give an emir more angry, because if you werent angry or dissatisfied you would use rica or geniş zaman (Ayran verir misin? Ayran verebilir misin? Ayran ver! Ayran versene!). I think you could use 'tatlı emir' indeed just to express you want someone to do something for you, without being rude or angry.
Also, it may express something like keşke.. like the example Erdinç gave about the traffic.. you wish the person did as you 'emired', but he does not.

Bana çikolata getirsene

Hepsi boş ya. O kadar gereksiz, bu konuşması!!

I got stuck at the 'making fun' part.. because I saw nobody making fun.

Now I will tell you something.. my father is a teacher too. He had a pupile whose last name was 'Vervelend' (Annoying). Ofcourse he wass teased with it, but that was no problem. One has to be mature enough to go along with such jokes as long as they are not made in a mean way. Anyway, I wanna come to this point. One evening my dad met his father, whose first name appeared to be Constant (Constantly). So when he introduced himself to my dad, he (had to) say this: Hello, I am Constantly Annoying.
Now if you ask me you can definetly joke about that and laughw hen you hear it. Believe me, that man does the same! I think one must have very long toes if you would make a problem of that..

Another one i know.. the last name of the mother of my sisters best friend is Chicken. She married a man whose last name was Rooster. Imagine her businesscard?
Ellen Rooster - Chicken.

Well sorry but if you dont laugh at such.. then your laughing muscles are made of steal! And laughing at it definitely doesnt make me disrespect her because it is a lovely woman.

'Dont walk with more than one person on this bridge', one scared man reads. He looks around and walks over the bridge but still the wooden bridge breaks down and the man ends up in the cold, deep water.. Bu nasıl oldu?

Een gewaarschuwd mens telt voor twee.
One warned man counts as two.