I would say: Did someone call me today? or Has anyone called me today? I guess "today" is used in the sense of "until this point in time" or "within this day", which of course has already passed.

Both answers are correct. "Today" has a variety of meanings. It can refer to a specified time period of today, e.g. work; school; etc. In such cases you can use the past simple. Think about it.

Did someone call me today? OR Was someone calling me today?

I think on the first option, the verb should be on the present tense since it has the 'did' word. So I think it has to be like: "Did somebody call me today?" or "Did someone call me today?" On the second one, I would rather say "Has someone called me today? With regards to the word 'today', I believe it is considered to be of past tense, same with 'this morning'. Well, these are just how I think it is, I'm not a native speaker of the English language. :)