I agree. As a native English speaker, there is little difference in meaning between using "impossible" and "not possible." Only, perhaps in the subtleties of whethe something may not be possible only in the moment as opposed to not being possible ever, or impossible. I am not familiar enough with the meaning of "nicht möglich" to say wch is the best fit, though. Nonetheless, in the case of this level of grammar, there should be a good explanation as to why these typically interchangeable words and phrases are not accepted.
I think, "Jedoch, es ist nicht möglich." is correct German, but it is unusual. The normal phrase is: "Es ist jedoch nicht möglich." I think, there is no other possible word order than these two.
You make a short break between "Jedoch," and "es ist nicht möglich." This increases the tension. So the "Jedoch," is emphasized.
Context is crucial. So please let me give you some examples.
"Ja, ich habe wirklich viele interessante Hobbies. Lesen, Schach, Musik hören. Wenn es möglich ist, Reisen, Autorennen, Tresore knacken. Jedoch, " und hier senkt sich sein Blick und er fährt leise fort: "es ist nicht möglich. Zur Zeit verbüße ich eine lebenslängliche Freiheitsstrafe."
(= "Yes, I really have a lot of interesting hobbies. Reading, chess, listening to music. Whenever possible, travel, car racing, cracking safes. However, " and here his gaze lowers and he continues quietly, "it's not possible. At the moment I'm serving a life sentence.")
The other word order is also possible if something follows:
"Ich singe gerne laut und ausgiebig. Von Beruf bin ich Taucherin. Jedoch ist es nicht möglich, beides irgendwie zu verbinden."
(= "I like to sing loudly and extensively. I am a diver by profession. However, it is not possible to combine the two in any way.")
I read "still" in that sentence as showing time-related meaning-- i.e., It wasn't possible earlier, and it still isn't now. You'd need just the right context and intonation for that to mean "however," and it definitely doesn't work in writing.
"Still, it is not possible" would be okay though.