Is the pronunciation just a really hard h or is it kind of a k sound at the beginning of herbatę?
"Herbata" is a feminine noun. You use "mieć" with Accusative.
herbata [Nominative] -> herbatę [Accusative]
(Co to jest?) - herbata; (Czego nie mam?) - herbaty; (Czemu się przyglądam?) - herbacie; (Co piję?) - herbatę; (Z czym jem ciasto) - z herbatą
In this specific example, is "I" dative?
I am really struggling with understanding "accusative", "dative", "nominative" and "instrumental". Any helpful tips would be greatly appreciated.
One thing that you can be sure of: the subject of the sentence is in Nominative, the most basic form. Here it is not visible, because "Mam herbatę" = "Ja mam herbatę", but "ja" is totally redundant as the subject is obvious from the verb's form.
No it's not. Using "having" for "eating" or "drinking" (as this is what "I am having tea" would mean) is a very specific usage, and a rather weird one, if you think about it. Assuming that it works the same way in other languages is quite risky. And it surely does not work the same way in Polish. "Mam herbatę" is just "I have tea". In my cupboard, for example.
Actually, it is not a weird usage at all - it is what native English speakers say all the time.
'What are you having?' 'I'm having tea' - not in the sense of having it stored in your cupboard but in the sense of drinking it now or shortly.
We often use 'have' where other languages tend to use 'take' as well; for example, when you talk about what you're going to order in a restaurant.
I guess I shouldn't have written 'weird' like this ;) What I (I guess) meant that it's an idiomatic thing that is used on a daily basis, sure, but if you think about the main meaning of the verb "to have", given the fact that the verb "to eat" is also a basic one, then (at least to a learner of English) it might seem quite strange that you use "have".
Anyway, the main point is that it's risky to assume that such an indiosyncrasy will translate directly into another language.
Although Ann's comment went rather the other way, trying to use "I am having", which means "I am eating", when Present Simple "I have" should be used.