self.BaseClass

When do I need to call self.BaseClass.PlayerInitialSpawn (exemple) and why?

Never, unless you want this to run.
Assuming you are deriving from the base gamemode.

Ok, so if I understand well, if I do self.BaseClass:HUDPaint() and I derive from sandbox, it will run the HUDPaint of sandbox?

Yes, I think so. Although it’s better to do (in your example) self.BaseClass.HUDPaint(self) - that way it works with your new gamemode table, and if you overrode any old values from the old gamemode it’ll use the new values.