I've noticed that many people here make use of copyrighted material for their games. (Stories, assets, etc.)

Isn't that illegal?

Can I use copyrighted material as long as my game will be distributed for free?

I don't work here, but here's what I've gathered:

This site (like most sites) allows users to publish pretty much whatever they want. It's an honor system.

Note the terms on the submitfile page:

By uploading a game here, you agree to the following:

• you are the author of this game, or have the author's permission to upload it for them
• if there is any copyrighted material in the game, you have permission to use it
• we can distribute this game on textadventures.co.uk and activelit.com

If a user publishes someone else's intellectual property and the copyright holder becomes aware of it, that's when the user could, should, and probably would get into trouble.

Copyrighted material is used under the "fan fiction" banner, which is quite a grey area. As long as you are not making money from it, most authors turn a blind eye, as it generates interest in their product (if you were to actually ask for permission, though, they would have to say no). Some authors have made specific statements one way or the other (JK Rowling has said fan fiction for Harry Potter is fine as long as you make no money for instance). I would suggest seeing how much fan fiction already exists. If there is a lot, you should be fine.

With regards to images and music, I am not so sure, but my impression is that they are more dodgy to use.

For Music & Images you could have a look at this topic. Eric provides these freely and is always updating them.

Also, aside from creating some of my own sounds, freesound.org is a great place to search for sound effects. They pretty much have everything for free and all any one there wants is an shout out if you have used their creation.

On that thought...
I'm computerizing a paper version of a DnD solo adventure to be played in Quest. I've made some changes, added bits here and there, and dropped a few. (Turned out to be harder than I thought!!!)
I would hope, giving credit to the original author for the source material would prevent any problems.
Currently it's over 500 rooms, or will be soon, and nearing 380Kb in size...
You would not normally visit all rooms, but it includes 6 characters, each with their own quests, so, it's 6 adventures in one.