Why did the ring choose Hal Jordan? I know that he is supposed to "overcome great fear", but there many other superheroes who hypothetically can use ring. Is there any proof that Hal Jordan is the most appropriate candidate from Earth?
First of all, it's important to note that Hal Jordan wasn't the only person on Earth who was qualified to receive the ring. When the ring was instructed to find a suitable candidate, it found at least two of them right away: Hal Jordan and Guy Gardner. There were also others on Earth (e.g. John Stewart) that would have qualified.
The ring selected Hal because, out of all the people that qualified, he was the most qualified person that was closest to the ring at that time.
Hal was qualified to bear the ring due to his overcoming his greatest fear: he was an Air Force pilot despite the fact that his father, also an Air Force pilot, died in a plane crash when Hal was young. The ability to face and overcome his greatest fear caused the ring to pick him out of those nearby.
It's probably true that other DC heroes would also qualify to wear the ring, but it seems pretty clear that the ring is at least partially sentient. It may not have wanted to select someone who already had a super-hero agenda, knowing that whoever it picked would have to leave Earth to patrol the entire sector as a Green Lantern. There is even a one-shot comic Batman: In Darkest Knight which shows several super-heroes (Batman, Superman, Women Woman, and The Flash) all picked as Green Lanterns. In that issue, Bruce ends up being removed from his position as Green Lantern because he focused all of his attention on Earth, while Sinestro ran amok across the rest of his sector.
Green Lantern Ring chose Hal because he was closest and probably best option at the time. We know he isn't the only Green Lantern of Earth but he was closes to Abin Sur's crash site. Also Many superheroes wouldn't qualify because of different personal traits as Batman/fear-Yellow Lantern Core, Superman/hope-Blue Lantern Core and Super girl was once Red Lantern.