"He speaks no Swedish" is not a correct translation for "Han talar inte svenska" since this sentence could mean two things.
Either "he" can't speak Swedish, or he is simply not speaking Swedish at the moment. Since it could be interpreted in both ways we only accept "He does not speak Swedish".
"Han talar ingen svenska" would be a perfect translation for "He speaks no Swedish".
I'm not entirely sure I get your question, but I'll try to answer.
I use inte in the sentence above since ingen means "no" as in "No Swedish".
You cannot use "inge", you have to choose between "ingen", "inget" and "inga" and they have the same meaning as "not any" (or "inte någ(ot/on/ra)" in Swedish).
Jag har inget vatten - Jag har inte något vatten - I have no water - I do not have any water.
Jag har ingen mjölk - Jag har inte någon mjölk - I have no milk - I do not have any milk.
Jag har inga vänner - Jag har inte några vänner - I have no friends - I do not have any friends.
Unfortunately, it's not as simple as "human v. nonhuman things". Every thing in Swedish (like most Germanic languages) is divided into different grammatical genders, which sometimes coincide with actual genders, but not necessarily. In Swedish, there's the common gender (combined from what used to be male and female genders) and the neuter gender (meaning non-male/non-female).
Common gender nouns are en-nouns (en mann, en hund, en sko, etc.), while ett-nouns (ett socker, ett barn, ett brev, etc.) are of the neuter gender. From those examples, you can tell that humans can be both en- or ett-nouns (en mann, ett barn), while non-human things can be either en- or ett-nouns, too (en sko, ett brev).
Both en and ett are translated as a/an, and English doesn't have the gender distinction, so it's kind of hard to explain the concept of gendered articles. I definitely recommend learning the en/ett article when you learn the noun, as it will help you when forming the correct adjective and definite forms.
I felt the link provided by Arnauti is useful, if you already understand the concept of gendered articles, but it's missing an introduction :)
The long a in Swedish is often perceived as an o sound by non native speakers, but with practice you'll get better at hearing the difference. Swedish has relatively many different vowel sounds, so we sometimes make distinctions that other languages don't. (the voice sounds ok here and I hear a clear long a sound in talar).
It's not acceptable in any region as a main clause, but it can be as a subclause. Jag förstår inte, eftersom jag inte talar svenska ('I do not understand since I do not speak Swedish') is a correct phrase (and you can't change places for either of the inte in there).
In clauses that are not subclauses or questions, the verb always needs to go in second place (the V2 rule), which means as a main clause, it must always be Jag talar inte….
There could have been a bug, or you made an error withour realising. If such an obviously correct translation is marked wrong for you again, please consider leaving an error report. That way, I can either see what went wrong or send it off to the developers as a bug.