Can you damage a bag of holding from inside

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.

No the bag will not burst, because Rope Trick is not an item

The key word in the caveat you cite is "item":

Placing a bag of holding inside an extradimensional space created by a Handy Haversack, Portable Hole, or similar item instantly destroys both items and opens a gate to the Astral Plane.

The two examples the text gives are also items.

However, the extradimensional space created by Rope Trick is created by a spell, not an item, and so does not fall under that clause.

There are many spells that generate extradimensional spaces (such as Mordenkainen's Magnificient Mansion)--if the designers intended the Bag of Holding to explode when it entered such spaces, they could have easily made the text refer to any extradimensional space, or used spells as examples.