No, it's isn't. 'Tomorrow' and 'today' ARE days, so it makes no sense to say 'Tomorrow's day'. If you want to specify that you mean 'day' as opposed to evening, you could say, for example 'During the daytime tomorrow'. Or if you are using the word 'today' in the broader sense of 'now', you could say 'the present day'.

No, but "this morning/ afternoon" or "tomorrow morning/ afternoon" are ok.

The others have answered so all I want to add is... you could say "tomorrow's daylight hours..." This assumes you want to say something about daylight hours! Tomorrow's daylight hours are few (because it's winter) Today's daylight hours were grey and grim!