I'll post my findings here in case anyone is looking for some answers, because it took quite a few days, and a lot of experimenting for me to figure this stuff out for myself (particularly the second bit).
Firstly, the sync button appears to now be called "Fetch Origin", which ought to be pointed out, as it is really not very obvious if you are trying to get GitHub to work for the first time.
Spoiler : Here's an image highlighting where the sync/Fetch Origin button is located on the desktop app. :
The URL required for the trial editor now follows the format https:// github.com/<username>/<repository name>.github.io/raw/main/<filename>.<filetype> (ignoring the <, >, and that one space after the https://, which I had to add to stop it from automatically turning itself into a hyperlink), thus, following the example in the original post (assuming that the judge has given his repository the same name as his username, as specified in the tutorial), the first example would become https:// github.com/udgey/udgey.github.io/raw/main/Courtroom Lobby.mp3 -- which is rather a long URL, and becomes even longer if your file name is complicated...
...But it just so happens that there is a trick to automatically getting the URL without too much trouble! Just go onto GitHub on your browser, and get up the information page about the file which you want to get the URL for - it should look like the image below:
Spoiler : :
I hope that this helps!