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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).
Today Was vs Today Is - English Language & Usage Stack Exchange
Today had been the worst day of my life. seems awkward, as today is not understood to mean on this day (the original meaning). In narrative, an event that is happened in the past is narrated as it is the present, as in: It is the 1st of April, 2006. Today will be the worst day of my life. Outside that specific context, I would write
Neither are clauses, but "today in the afternoon" is grammatical (adverbial phrase of time), while "today afternoon" is not. I would also suggest "this afternoon" as a more succinct and idiomatic alternative to "today in the afternoon".