Is it possible to use this phrase to ask how a specific thing is going? For example somebody just started a new study and you want to ask how that's working out for them. In English, I would feel comfortable asking 'And how's that going?'. This translation is not accepted here, but I wonder if in Czech you could use the phrase 'Jak to jde?' or 'Jak to de?'
Actually, you can. It is what many native English speakers in the U.S. say, half jokingly, instead of How is it going?. It is far more natural than the We buy a new house! for Kupujeme nový dům! that you were justifying by saying that we are learning Czech here, not English.
It is not the main translation and was only suggested to you in response to something closer to it than to the How is it going? main translation. I suspect your attempt was How does it go?, which again uses the simple present awkwardly, plus it even has a very different meaning, a question about how a song or saying "goes" when we cannot remember. That would be something like Jak je to? in Czech.
"How goes it" is certainly something you can say, but to suggest that "many native English speakers in the U.S. say," is just flat out wrong. We say, "how's it going?" way more often than not. You're more likely to hear: "What's new?" "How are ya?" "What's going on?" "What's up?" and a host of others before you hear "How goes it?"
Having said all that, you're absolutely right about "how does it go?" That shouldn't be acceptable. Sure, it's grammatically correct, but you would only say that to ask about the melody of a song (at least this is definitely the most likely scenario). Whereas asking someone "how goes it?" (while corny, outdated and frankly very lame) is asking about their life.