3 “Earlier today” is a totally correct way to refer to a point in time between the beginning of the day and the current time. Because it refers to a moment in the past, it can be used with the past tense, as you did in your example.
Today means "the current day", so if you're asking what day of the week it is, it can only be in present tense, since it's still that day for the whole 24 hours. In other contexts, it's okay to say, for example, "Today has been a nice day" nearer the end of the day, when the events that made it a nice day are finished (or at least, nearly so).