I need some help for this, because it might be just me or some obscure settings somewhere which I missed.

When osu!player (I'll coin it that to let you all know I'm referring to osu! as a music player) finishes a song, it doesn't automatically move on to the next song.

In song select, when the song ends, it doesn't restart.

Happens on my side as well. I don't know what triggers this, so I can't reproduce...

this usually happens when i had a song in "song select" (or played a song) and then i go back to the main menu then it gets stuck there, correct if im wrong, but im not 100% sure on it, i am 100% sure on the fact it's happened to me too.

I never had the "doesnt restart" issue in song select tho.

Same problem here, but I can't find the method to reproduce it.

How often does this happen?

It just happened. I'm probably going to reproduce it trying the following cases sometime today.

1) Start game -> Let song finish without going to song select. Nothing
2) Start game -> Go to song select -> Let song finish in main menu. Nothing
3) Start game -> Go to song select -> Let song finish in song select. Nothing
4) Start game -> Play song -> Return to song select -> Let song finish in song select. Nothing
5) Start game -> Play song -> Return to main menu -> Let song finish in main menu. Nothing
6) Start game -> Play song -> FunToggleWhatever -> Mess with anything in the box -> End game -> Return to main menu -> Let song finish playing Positive, reproduceable

Apparently, while doing case 6, the box can also appear in the middle of play. Is this intentional?

It's probably highly dependent on which mp3 is playing (I'm guessing something weird about the encoding), so please test with different beatmaps and see if one consistently reproduces the issue if possible.