Re: the first part, I'd certainly not put all of it down to Werner individually, but I do think it's fair to say his passing abilities in tight spaces have been improved under Nagelsmann. Looking at expected goals /assists in that context would be misleading, but I think looking at key passes at least is illustrative since these are most likely to come from tight spaces in and around the box.
I agree that Werner didn't look great at the last World Cup, but to be fair that German team was a trainwreck from the jump - I don't rate Low at this point at all quite frankly. It's also worth pointing out that Werner was only 22 with (I think) only 12-15 caps when he went to the WC. I'd also argue that his ability to play as a false 9 has really been a focus for Nagelsmann this year - he's said that he thinks the best way to use Werner is to have him drop deeper towards midfield so he can build a head of steam as he runs towards defenders. While he's mostly been used in a pair or off the left this year, Nagelsmann has also used him in this sort of withdrawn role. I do agree that this would solve a lot of potential issues for Low; we'll see if he gives that a go moving forward now that Sane is fit again.
You're right that my use of "LW" was incorrect; I should have described him as more of an inside left forward. My mistake. Aubameyang is a very good comparison.
I would just take everything with a grain of salt when you look at a player's stats and take out of context that he plays under a coach that organizes his teams extremely well and gives him a free role in that framework. It's part of the reason why people have been so disappointed with players signed from Dortmund. His passing stats (kp/90, xA/90) indeed look good for a striker, but for what it's worth my personal impression is that while he can take part in combinations via layoffs and the likes he doesn't have particularly great playmaking qualities, if I was bothered I'd check how many of them came from squaring it to a team mate after Leipzig flooded the opposition area.
His troubles in the national team didn't end with the world cup, if anything they have gotten worse, with Löw exploring other options upfront, such as Freiburg's Waldschmidt, Schalke's Uth, Reus, Gnabry and Sane. During the Euros qualifier he's started just 3/8 matches, finished none of them, been subbed in towards in the end in two and fully benched in three. Even Löw is not such a big idiot that he keeps overlooking a top class no9, so he can experiment with midfielders upfront.
What you're quoting about Nagelsmann isn't really the role of a false 9 (in the sense of someone nominally playing upfront, then dropping deeper to pull away defenders and maybe do some playmaking Messi style), he's letting Werner attack from different(LW) or deeper angles, so he's harder to mark and can run at defenders with his pace. But at the same time Schick or Poulsen always remain to lead the line for him.
Which is really why I'm wondering how Chelsea plans to balance all this. Ziyech is basically a no 10, Werner is basically a second striker who demands someone like Giroud (as opposed to Tammy) to play, Havertz is basically a second striker as well, Pulisic has been best when cutting inside from the left too. I think you would need truly amazing fullbacks to make this work efficiently and even then there might be balance issues.