I think I didn't watch Naruto carefully because I can't understand one thing: After the third Hokage died, the fifth was chosen. Why did they skip the Fourth Hokage?
After Sarutobi Hiruzen (the third hokage) stepped down, Namikaze Minato (Naruto's father) became the fourth hokage. Minato died during the kyuubi attack during which it was sealed inside Uzumaki Naruto. Konoha now lacking a hokage, they chose to bring back Sarutobi into office.
So they didn't skip the fourth hokage, they just reinstated the third one after the fourth...
There was a 4th Hokage when the 3rd Hokage stepped down.
But then the 4th Hokage died prematurely, meaning that the 3rd Hokage had to take up the position as Hokage back (becoming the fifth).