"Tą herbatę" is incorrect. However, this slip is so popular among Polish people that it is acclaimed as the other correct one in the collocquialism. But it should not be used in Duolingo, I think.
I was taught differently. The following vowels I type, please assume they have tails - my keyboard has not been set up yet to type those Polish vowels. Anyway, I was taught by someone whom speaks Polish very well, grammatically. She said there is not such word as "te" (E with a tail). Accusative would be Ta Herbate (both the A at the end of TA and the E at the end of Herbate will have tails). It would never be Te Herbate (both Es at end of both words have tails). The word does not exist and I cannot find it in my dictionary, either. So, shouldn't "Te mysz" above be "Ta mysz" (both vowel endings have tails). In the meanwhile, can someone teach me how to adjust my keyboard so I can type the Polish vowels with tails? Many thanks.
Depending on whether your device is apple or android, there's different ways of adding keyboards.
For apple devices: Settings -> General -> Keyboard -> Keyboards -> Add New Keyboard
For andriod devices (I have a samsung phone): Settings (this might be in "Apps" in the bottom right) -> General Management -> Language and Input -> Samsung Keyboard (I'm assuming there'd be an equivalent for other phone makes) -> Languages and Types -> Add input languages
From there you scroll to the language(s) you want to add! You can change between keyboards by swiping left and right on the space key for android and press and hold the world icon/smiley face for apple. (Sorry if other devices have a different route, most of them should have a similar way of getting to languages at the very least.)
Although for easy access you can press and hold on any letter key and it should offer variations of that letter (including ę and ą for e and a respectively).
Ę or ę is a letter/vowel in the polish language. However I myself have never seen the word Tę before.
It should be 'tę herbatę', as both 'tę' and 'herbatę' are accusatives of 'ta' and herbata'. 'Tą' is instrumental, and, just like Viersch described, shouldn't be used with accusative 'herbatę', but it's pretty popular to mix these two in coloquial speech. But for you as a learner, it would be better for you not to do it, to avoid developing bad habits (my ears bleed when I hear 'tą kobietę' from natives).
Answering the 'tę mysz' part - mysz is a feminine noun that in nominative ends with a consonant 'sz', thus it doesn't change it's form in accusative. Simple (or hard) as that, you just have to remember it :)
I thought that ,,mysz'' in accusative form should get ę, since ,,mysz'' is feminine gender. Am I wrong?
Well, that's true that it's feminine, but it looks masculine and behaves... well I don't know if masculine, so let's say irregular.
There's a colloquial word "mysza" and then the Accusative would be "myszę" indeed, but that's sure not the standard language.
Masculine - in nominative case it'd be 'ten', in accusative - 'tego'. Neuter - in nominative case it'd be 'to', in accusative - also 'to'.
In example: Widzę tego męzczyznę (I see this man). Widzę to dziecko (I see this child).
So for "then" it would be "tamtego" for maculine and "tamto" for neuter? Thanks for your answer!