Kakashi became a Genin at the age of 5 and Chunin at 6.
Obito became a Genin at the age of 9 and Chunin at the age of 11.
It was around the time when Obito became Chunin that Kakashi was promoted to Jonin.
Obito then "died" at the age of 13, which was almost two years after his promotion to Chunin.
From this thread on narutoforums (emphasis mine):
It is impossible for both Kakashi and Obito to have been 13 during the Kakashi Gaiden, because there is at least a confirmed 4 year age difference between Kakashi and Obito when they were confirmed to both be Genin teammates during the Chuunin Exams. They entered the academy the same time, and they all graduated the academy together as teammates. Kakashi was at least 4 years younger than both of his teammates at all times. There is no possible way that Kakashi was a 13 year old Jonin during the Kakashi Gaiden.
At this idea that he was 13 isn't even supported in the databooks, either. It's time we all got our facts straight. If Obito died at 13, then Kakashi was most likely 9 during the Kakashi Gaiden. It is impossible for these two teammates to have ever been the same age.
Kakashi was 13 during the Kyuubi attack on Konoha. There is a 4 year difference between the events of the Kakashi Gaiden and the events of the night Naruto was born.
Also, from the talk page of Obito Uchiha on Naruto wiki
Obito-Kakashi timeline is based on the anime mostly and pre-chapter 599 manga/databook stuff. Kakashi graduated the academy as 5 years old, Rin and Obito as 9. Obito is 3 years 7 months older from Kakashi. They all took the same Chunin Exam, with Kakashi having managed to become a Chunin in it at the age of 6, while Obito age 10 failed. Obito trainer hard, took the next exams and became a Chunin at 11. Obito got crushed by boulders as 13 years old, that means Kakashi was 9-10 at the latest in the Gaiden. Kakashi is now 31 so Obito died as 34.
As for the youngest Jonin part, no one really knows how many official Jonin are there. If I remember correctly, Itachi wasn't officially declared as a Jonin. He got into the Anbu after he became a Chunin. So Kakashi might be the youngest Jonin. There is no definitive answer.
Edit (courtesy of @Merrice Henderson):
There is evidence that Kakashi made Jonin the same year that Obito and Rin made Chunin. This is an exert from Obito's wikia page:
Eager for Rin's attention, Obito trained relentlessly, eventually rising to the rank of chūnin himself. His excitement was short-lived as Kakashi soon afterwards became a jōnin, once again earning Rin's praise and Obito's resentment."
So, Obito was confirmed 11 when he became Chunin. If we go based on the theory that Kakashi is 4 years younger (which I am more inclined to believe), then he was 7 when he made Jonin. That would effectively make him the youngest Jonin of all time. If Kakashi was the same age as Obito and Rin, there are a few years that need to be accounted for, like:
"Who was he teamed with after graduation while Obito and Rin were still in the academy?"
"What happened to Kakashi's other teammates, or was he on hiatus for that time?"