His "best friend" and teammate Obito was a Uchiha and bequeathed the eye to Kakashi upon his death. His other teammate, Rin a medical specialist, installed the eye. Kakashi had just been promoted to jounin and it was his first jounin mission. He was 13 years old.

Besides looking cool, it's less drawing for the creator and his assistants and the other animators.

Practical reasons - it might be a protective filter against poisonous gas or allergies; prevents reading lips; would normally hide a shinobi's identity, but in his case the hair is a dead giveaway! A mask also hides expressions so a shinobi won't give away his intent. There are other shinobi who wear a similar mask all the time. I don't think it's a separate mask, but a turtleneck extension of his shirt. He also doesn't eat in public very often. This might also be a precaution from being poisoned. So all woman don't fall immediately in love with him!

His nonconformist personality (or as Gai puts it hip/modern/revolutionary) indicates he doesn't care what people think. He may also do this on purpose to develop a non-approachable persona. What girl would really want a guy who reads books like that in public?

When Kakashi says his best friend's name is on the shinobi memorial, it can also be translated as best friends' names. However, later we find he is mainly concerned with honoring his best friend, Obito.

In addition to having passive-aggressive tendencies and being unconventional/nonconformist (these personality types tend to be late all the time), we find in episode 80 that he comes to visit the shinobi memorial often in the morning. When he's there, he thinks about his past and becomes guilty/depressed thinking about what a fool he was back then. He blames himself for his friend's death.

The girl is Rin (no surname) a medic-nin with a crush on Kakashi.
The boy with goggles on his head is Uchiha Obito.
Yondaime (no actual name) was their sensei before he became hokage.

However, Kakashi graduated the academy and became a genin at 5 years old. His teammates were the same age and were chuunin when he became jounin, so maybe they were geniuses, too.

Okay, let's just do Gai who claims he has 50 wins vs. 49 losses against Kakashi. That's a lot of fights over 26 years, maybe they started as toddlers fighting over a rattle! These fights were probably training sessions where they try out new techniques. Since Kakashi became a genin, chuunin, and probably jounin, before Gai, it is unlikely that any of the fights were in a formal setting where they would go all out. The "fights" also include non-combat challenges like rock-paper-scissors (see ep 101).