You can place "vandaag' anywhere as long as the inflected verb is in second place (the second constituent) So options are:
Ik heb geen tijd vandaag.
Ik heb vandaag geen tijd.
Vandaag heb ik geen tijd (inflected verb comes 2nd and the subject moves to the spot after the verb)
(NB this only counts for main clauses, things are different in subclauses)
That thing about the continuous form isn't a rule, just a generalisation. Normally it is the better version, or may be the only correct version. However there are times when it is completely wrong to translate using the continuous form, as in this case.
"I am having no time today" doesn't make any sense.
I think it is because some verbs in English, like 'to have' and 'to see' don't often use the continuous tense. Like if you were holding two apples, you would say 'I have two apples',(present simple), however you would say 'I am holdING two apples'(continuous) Saying 'I am having two apples'(continuous) would sound weird. Some exceptions I can think of are "I am having a bath' 'I am having a sandwich for lunch', so maybe you use 'have' for possessing something and 'having' for performing actions, I'm not 100% sure