Some note taking apps will play audio files.
Look at Notability.
Notablity WILL save and play mp3 files.
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Also, a web browser with a download manager will only download the file inside of the web browser and not to other apps in iOS.
To get a file to an appropriate app you would need to look for the Share feature in the web browser app (either by the name "Share" OR a symbol that looks like a square box with an up facing arrow through it) that will bring up a window that will allow a file from the web browser to be copied to an appropriate app of your choice.
There is no way to save any outside music files to the iOS Muisc app. This app is for iTunes content from either a computer OR paid and downloaded from the iOS iTunes store.
You need to find apps (free or paid) that will store and play different types of audio files.
Notability is just one app that can do this.
Also, look at all of the various mp3 player apps that are available for iOS.
You may or may not find a free mp3 player that will work, but maybe pay solutions that do.
All of these apps maybe NOT free, but solutions for playing outside, downloaded mp3 audio files exist in the iOS App Store!
Good Luck!