Why was there a real bullet on the set to begin with? And who...

Yes, there were real bullets on the set. Apparently what happened was they sent a prop guy to buy real bullets and then do a DIY conversion of those bullets into "blanks".

Here's a description (bolding is mine) from Wiki (yes it's Wikipedia but the story is well documented and likely to be as accurate in this case as any other information)

A previous scene using the same gun had called for inert dummy cartridges fitted with bullets (but no powder or primer) to be loaded in the revolver for a close-up scene; for film scenes which utilize a revolver (where the bullets are visible from the front) and do not require the gun to actually be fired, dummy cartridges provide the realistic appearance of actual rounds.

Instead of purchasing commercial dummy cartridges, the film's prop crew created their own by pulling the bullets from live rounds, dumping the powder charge then reinserting the bullets.

However, they unknowingly or unintentionally left the live primer in place at the rear of the cartridge. At some point during filming the revolver was apparently discharged with one of these improperly-deactivated cartridges in the chamber, setting off the primer with enough force to drive the bullet partway into the barrel, where it became stuck (a condition known as a squib load). The prop crew either failed to notice this or failed to recognize the significance of this issue.

In the fatal scene, which called for the revolver to be actually fired at Lee from a distance of 3.6 - 4.5 meters (12–15 feet), the dummy cartridges were exchanged with blank rounds, which feature a live powder charge and primer, but no bullet, thus allowing the gun to be fired without the risk of an actual projectile. But since the bullet from the dummy round was already trapped in the barrel, this caused the .44 Magnum bullet to be fired out of the barrel with virtually the same force as if the gun had been loaded with a live round, and it struck Lee in the abdomen, mortally wounding him.[9] He was rushed to the New Hanover Regional Medical Center in Wilmington, North Carolina, where he underwent 6 hours of surgery. However, attempts to save him were unsuccessful, and Lee was pronounced dead at 1:03 p.m. EST. The shooting was ruled an accident.

The metal fragment that killed Brandon was not a typical bullet, but a blank, self-made by the crew. They didn't have money in the budget to buy any more blanks, so they made their own (pretty stupid.) The blank didn't fire correctly and got stuck in the barrel. When the second blank was fired, it propelled the stuck blank out of the barrel, like a real bullet.

Also at fault was the prop master who didn't check if the gun barrel was clear before the second shot. If you can find a copy of Unsolved Mysteries episode about this, it explains it really well. They said ultimately no one was charged because the chain of mistakes involved too many people.

they left the primer in the gun , a deactivated blank was left in the gun so when fired, the metal penetrated his abdomen and he basically bled out

How Brandon Lee actually died on set of The Crow

https://www.youtube.com/watch?v=K3cE-Azcw44

These are eyewitness accounts from those who witnessed Brandon's death on set, as well as an experienced Armourer showing exactly what they did with the gun on set in those days, so all the conspiracy crap can be put to rest, this is the proof.

It was a squib, not a real bullet.

A dummy doesn't have a live primer. It doesn't take a lot of technical training to understand how cartridges work. It's really pretty simple. It's amazing that bullshit like this happens on movie sets.