"Have you tried the Taiwanese bubble milk tea before?"
I disagree with you here. If they're looking at a menu, for example, then I can see how the 'the' is completely appropriate.
There could be a "the" if you're talking about the bubble tea on a specific menu, rather than in general.
No, with the caveat that the nuance changes a bit, making it seem like the full name of the thing is 台湾珍珠奶茶.
To my mind the menu-item scenario that's been suggested, if specifying that it's actually Taiwanese bubble milk tea, would in fact leave out the "的". Perhaps that's partly what's swaying me to interpret the given Chinese as a general question, and not a reference to a menu item (and therefore to feel that "the" isn't a particularly appropriate addition to the translation).