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. :)