Okay, this is really silly, but one might argue that there could be situations where the translation would actually be used, whether or not it's "technically" correct. For example...
-- You haven't seen František for years, so you thought he was dead... and, suddenly, you bump into him on the street. "František, you exist?!?!?"
-- You live in an area where liberties are routinely taken with the English language. You haven't heard from František for a while, and then you run into him at your favorite bar. "Yo, František... you exist, man??"
In any case, "František, do you exist?" was accepted.
Actually, I think a better situation could be used.
You're walking around Prague hearing people talking about a so-called "František", and you don't believe he exists.
And then you bump into František and you're like, "František, you exist?"
(Idk if František would take offense to that)
In short, asking questions like "You read?" or "He snowboards?" only make sense when there is disbelief.
As I understand it, there is no "do" block in the word bank because "do" is not part of the main translation, i.e., the one at the top of the discussion page (as of 26 Aug 18). Since translations with "do" are accepted, I assume there's a reason (slang? silliness factor? keeping us on our toes?) why the main translation doesn't include "do." If that should change, a "do" block should magically appear.
Native English teacher input. You are all correct in your comments and hypothetical scenarios except for the punctuation.
In English this could exist as a statement. Bonehead Bass' situation (of incredulity at finally meeting F) could certainly happen and the exclamation is an example of a common remark. However, the words would be punctuated with just an exclamation mark, not also a question mark.
If you want to phrase it as a question, you need the extra word 'do'.
To know from the horse's mouth whether František exists, for example?
But that's not the point, the point is that it's as good a sentence to practise Czech as any other. In fact, I find those borderline nonsensical phrases such as “to není tvůj medvěd” great—there seems to be didactic value in the surprise effect.