Learn English – Does a gerund require a preposition after it

It is not just the preposition to that takes the gerund, but the construction "be adjective to":

Locking him up without food or water is tantamount to killing him.
Knowing your budget is important to successfully planning your holiday.

There are lists of verbs that take an infinitive or a gerund, like here, but they are seldom complete.

The gerund is also after verbs like "looking forward to":

I look forward to meeting you
He looks forward to getting to know her better.

Let's have a look at another example sentence to see if there is a difference between "be adjective to gerund" and "be _adjective infinitive":

Sharp knifes are important to cooking.
Sharp knifes are important to cook.

In the first sentence, I state that an important part of the action of cooking consists of sharp knifes. The sharp knifes can be seen as a property of cooking.

The second sentence is different. Well, there is the possible confusion that I mean I should cook my knifes, for one. That makes little sense. Even if we ignore that, I would parse the sentence more like:

If I am to cook, it is important I have sharp knifes.

So if I have no sharp knives, I won't start cooking.

Somehow, the sentence gives me more of a specific feeling, whereas the cooking variant is a general observation.

Now, in your original sentence:

changes are integral to achieving improvements.
changes are integral to achieve improvements.

The first sentence makes changes an (integral) property of "achieving improvements" in general.

The second one seems to presuppose we have a specific situation where we want to achieve improvements, and in order to do that, we need changes.

Strictly speaking, the sentence is incorrect because listening of playlists is ungrammatical (unless the playlists are doing the listening). Since categorization takes of and editing takes of but listening takes to, you have to write this:

Some websites allow categorization of, editing of, and listening to playlists online.

This is grammatically correct but it sounds very clumsy. If each noun took the same preposition, you could use the same preposition for all three:

Some websites allow categorization, editing, and playing of playlists online.

This is grammatically correct but it sounds even clumsier because of the repetition of play. People would rather make a subtle grammatical error than write a sentence that sounds this clumsy.

The fact that the first two of the nouns take of probably led people to ignore the incorrect listen of for almost ten years now.

Another “fudge” solution is to choose the preposition to agree with only the nearest noun even if it disagrees with all the others, known as “proximate agreement”:

Some websites allow categorization, editing, and listening to playlists online.

There is, however, a better way:

Some websites allow users to categorize, edit, and listen to playlists online.

This is clearer because the users are mentioned explicitly, and the nominalized verbs are replaced with plain old infinitive verbs. The preposition to only agrees with listen, but that's OK: categorize and edit are transitive verbs, which take an object without any preposition at all. So, to connects only with listen and there is no disagreement with categorize and edit. So, this version has perfect grammar as well as greater clarity.

By the way, many gerunds do take of. For example: editing of playlists, feeding of animals, planting of gardens, singing of songs, etc. Also, gerunds normally function as nouns. In the original sentence, editing and listening are objects of allow, just like categorization.