When I started the game I was prompted by my firewall for giving the permission to the game to go online.
Being it a singleplayer game with no updater, why does it try to go online in the first place?
Anyone has a clue about this?

Same here. The firewall advise after launhing the game.

Same thing here too. I first thought it was an update thing - but as there is no auto update function integrated, I don't know why it wants to phone home.

Same here, I buy from GOG so that I don't have to deal with DRM and this seems like it could be some sort of check. I denied it for now but would like to know if I'm missing any "features" or not. Haven't had a chance to play it yet, just launched it to make sure it ran.

I think it's is mandatory to know why the game try to connect somewhere and should be a GOG top priority clarify this. If there's no reason to do that, and I believe there's not at least for us, this should be patched away.

Maybe want to connect to the cloud to save the game .Sarcasm mode on.

Send a support ticket to GOG.com and see what they say. they usually don't have anyone in the office on weekends. But they 98% of the time will replay in the same day.

Sent ticket, got reply. They say to me to ask the developer. While I agree that they might should know more about this, I think that's not a great support answer. GOG sell the game, so they should at least try to find out more.

Got an official reply from GOG support.
Ethan Carter runs on Unreal Engine 3. This engine, no matter the game, always tries to open up a port (but not listen on this port) on a user's system. Tower of Guns, another UE3 game, does the same exact thing. In short, it is a completely harmless quirk of the engine and the only way to stop this game from doing this would be to request as such from the makers of the engine it actually runs on.

In summary: it is a Unreal Engine 3 quirk that we cannot change. You can tell your firewall to block UE3 and you can still play your game.

Thanks for looking into it and reporting this to us MIKO!