Jsyn is close.
1) A dropped firearm MAY fire, particularly old school revolvers with the firing pin on the hammer if it falls on the hammer. However, it is much more common with long guns, which often have a free floating firing pin. The firing pin does not have a block, such as a drop safe handgun has, and as the name implies it floats freely in the bolt. The AR-15, the Remington 870, and many others all have a free floating firing pin and can discharge if struck hard enough to give the pin enough inertia to touch off the primer. This is why these weapons shouldn't be stored with one in the chamber, and also why military primers are traditionally harder than civilian ammo primers.
The second common way is a "slam fire". This is where the firing pin gets stuck with the tip sticking out of the bolt face, so when the bolt is closed the firing pin strikes the primer. A bolt action rifle, for example, with a bit of debris in the firing pin channel. You work the bolt rapidly, and the gun discharges. A pump shotgun could discharge when you work the pump for the same reason. This can cause a 'run away gun' in a semi-auto or auto, where the gun will fire until the magazine is empty as the bolt fires every round as its chambered.
Lesson: When closing the action on a firearm, make sure its pointed in a safe direction, as its possible to discharge the weapon without touching the trigger as the action closes.
2) Modern thinking is let it drop. Modern handguns are drop safe and the possibility of a discharge is much smaller than the possibility of accidentally pulling the trigger trying to catch it.
3) While that would work with a bolt gun or inline muzzle loader many guns cannot be viewed from the action end. A chamber flag is the preferred method. Once the gun is empty, both chamber and either magazine removed (external) or empty (internal), a chamber flag is inserted. This is a small device that is inserted in the chamber, it will not go in in there's a cartridge there, and also physically blocks the firing pin. This renders the weapon mechanically incapable of firing, even if there was ammo in the magazine, as it can't be chambered and the firing pin can't hit it. Fiber optic chamber flags also create an easy way to illuminate the barrel, simple shine a flashlight on the end that's outside the chamber and it will light up the barrel.
4) Yup, anyone who sees something unsafe has the authority and the duty to call a ceasefire. Most ranges will eject anyone who fails to honor a ceasefire, if not ban them.
5) Yes, with the purpose being to catch the bullet if a slam fire were to occur. Sand barrels are often literally barrels of sand, but can also be a block of polymer in a steel can, a stack of phone books in a milk crate, etc. The purpose is when loading and unloading a weapon capable of slam firing to put the muzzle in the sand barrel just in case.