The rules as written say only that a bag of holding can be pierced from the inside. They don't say how hard a bag is to pierce. In other words, the rules don't actually answer this question explicitly... But that doesn't mean we can't work out the answer.
After all, we know that a bag of holding is a bag. More specifically, bags of holding are traditionally described as being sacks, though backpack-style bags of holding have also appeared in large numbers. Sacks are usually made of sackcloth, and backpacks are usually made of leather - and since we know what materials bags're usually made out of, we know how hard they are to cut: As per page 175 of the core rules, cloth has two HP per inch of thickness and 0 hardness, while leather has 5 HP per inch of thickness and 2 hardness; and as per page 459, magic items take damage as nonmagical items of the same sort unless otherwise noted: Since a bag is presumably less than an inch thick, it wouldn't be too hard to stick a knife through.
How likely is that to happen, though? If you place a knife carefully onto sackcloth instead of just tossing it in, the chances of it piercing the bag are almost zero, right?
Well... we know that a bag of holding has a maximum capacity measured in weight, and that it can carry up to that weight without trouble. A bag of holding is therefore unlikely to break just from the weight of stuff inside it. That implies that it's the cutting edge of an object, and not the amount of force behind it, that makes carrying a sharp object in a bag of holding dangerous. (We probably could have guessed that, but it's nice to have it confirmed.)
We also know that items place inside a bag can shift around. The rules for bags of holding don't actually say that their contents move when the bag is moved, but it's a common sense assumption for normal bags, and that means it's probably a safe assumption for magic bags as well. Even if your GM rules that the contents of a bag of holding don't shift around when the bag is closed, he or she will probably agree that they shift around when you stick your arm in to dig around for something - and that means that your hypothetical dagger could potentially be pushed into a position where it risks damaging the bag without your being aware of it. Uh-oh.
There's your conclusion: It's possible to put sharp objects in a bag of holding without damaging the bag... But there's risk involved.
But wait, you say, why is it so many people are adamant that you should never put a sharp object in a bag of holding? Well, a bag of holding is an expensive item that's used to store other expensive items; Many adventurers are understandably reluctant to risk losing what amounts to all their worldly wealth for a stupid reason. That, I suspect, is where the aversion to putting pointy things in a bag comes from.
Fortunately, even if your GM disagrees, it's not a real problem. Just buy a box, and put the box in your bag; Then, put sharp things into the box. It can even have rounded edges, if you like.
Best Answer
Option 1: Bag breaks
This is up to a DM ruling, but it's possible that falling from such a great height, especially full of items (the bag always weighs 15 pounds, after all) could deal enough bludgeoning damage to the bag so that it ruptures. DMG 153 states,
Thus, your gnomish friend is now somewhere on the astral plane.
Option 2: Gnome takes damage
While I thought that the bag's opening is really a portal to some extradimensional space, it's actually just bigger (DMG 153):
This passage, according to my reading, states that while the bag has extraplanar properties, the items inside the bag are still inside the bag, and not tucked away in some extraplanar space. It's just that magic makes the space bigger than it would normally be.
Thus, because being inside a bag doesn't protect you from fall damage [citation needed], your gnome friend takes the 12d6 from falling.
Generally, this plan doesn't seem to result in the best outcomes for your gnome friend.
Compare this to the Portable Hole, which explicitly states that it opens to an extradimensional space (DMG 185-6):
Because that space is actually on a different plane, a creature hiding inside a portable hole would probably not take damage from the fall (unless the item was damaged in the fall somehow).