I've been gathering the info for awhile now, and I've finally got the evidence that Hiruzen was the youngest Hokage. So then, I'll begin.
We know that Konoha was founded 60 (63) years ago, and Hiruzen had died at the age of 68. That means he was 8 years old when Konoha was founded.
The Academy was founded by Tobirama, and therefore after Hashirama's death. This means that Hiruzen was trained by Hashirama/Tobirama before the Academy was built. That would mean Hiruzen was trained at the age of 8 at the earliest.
Hiruzen graduated the academy at the age of 12.
That would mean that Hashirama died within 4 years of finding Konoha.
Quote:
Originally Posted by DB1 Tobirama Entry
The Nidaime is the Shodai's younger brother by blood, who inherited his brother's dying wish and was called "Hokage" after the Shodai died. He endeavoured in the founding of the Academy and the establishment of the village's organisational system. However, shortly thereafter the Shinobi World War broke out. He entrusted the wishes of the suppression of war and the prospering of the village to his successor and died a noble death.
This information tells us that shortly after Tobirama created the Academy, the [Second] Shinobi World War broke out.
Spoiler:
Its unknown when exactly the Academy was created, but we know that it was created in less than 4 years after Konoha was founded (because Hiruzen was 8 when Konoha was founded, and 12 when he graduated the academy).
Since the Academy was just recently founded by Tobirama before the Second Shinobi War, and Hiruzen graduated at 12, I'd say that the Second Shinobi War occurred when Hiruzen was 12 or 13 at the latest.
Tobirama died during this war (not necessarily on the battlefield) and entrusted Hiruzen with ending the war.
Quote:
Originally Posted by DB2 - Jiraiya Entry
Relationship with the 3rd
The 3rd is not only the ruling Hokage of the village but also at the same time a teacher to Jiraya. Because of the teacher and student relationship, a special connection exists between them. To his teacher who met a heroic death during his fight with Orochimaru, what might have been his(Jiraya's) last words?
Link
Hiruzen was the (3rd) Hokage while he was training the kid Sannins.
We know that Oro/Jiraiya/Tsuanade all graduated the Academy at the age of 6. That means during this time, the Sannins were 6 years old and just taking the Bell Test.
The Sannins were 51 back in Part 1 and Hiruzen was 68. So, 68-51 = 17. That means that Hiruzen was 17 when the Sannins were born. He trained them when they were 6.
So, add 6 years to Hiruzen's age of 17, add you get 23. That means, at the maximum, Hiruzen was 23 when he became Hokage.
Finally, factor in the scan below, and the age [range] Hiruzen became Hokage becomes quite clear.
Scan:
Minato's Hokage Age Range
Alright, I've got a range on Minato's age of becoming Hokage.
The Third Shinobi World happened over 10 years ago before Part 1.
During this war, Minato was a Jounin and Hiruzen was Hokage.
We know that Minato was Hokage at least 16 years ago before Part 2 because he was the Fourth Hokage when he sealed the Kyuubi when it attacked. That means (at least) 13 years before Part 1, Minato was Hokage.
Based on this information, we can conclude that the Third Shinobi World War happened over 13 years ago.
Kakashi was 13 during during this war.
5 years ago from Kakashi Gaiden, Sakumo killed himself. That would make Kakashi 8 at the time.
Moving on with further info.
We can conclude that Sakumo killed himself during the Third Shinobi World War because the Fire Country suffered great losses at due to Sakumo's failure and because Sakumo's mission was to sneak into the enemy's land. Konoha didn't have any enemies during the time of peace, until the Third Shinobi World War.
That means that the Third Shinobi World War happened about (or over) 18 years ago.
Kakashi was 26 at the beginning of Part 1. 26-18 = 8.
So, this tells us (and confirms) that Kakashi was 8 when the Third Shinobi World War occurred, as well as when Sakumo killed himself.
Since Kakashi was 8 when Sakumo killed himself, then he was 13 (minimum) when the Kyuubi attacked Konoha. Because at the start of the series, Kakashi was 26 and the attack was 13 years ago before Part 1. That means that shortly after Kakashi Gaiden, the war ended and Minato became Hokage.
So, the key to finding how old Minato was when he became Hokage, is to find out how old he was during Kakashi Gaiden.
More info later (currently looking into Minato and Nagato's age differences).