fontainbleau is perfectly credible, bi
Verdun to Paris certainly, and Fontainebleau is quite possible, but why Flanders? That would make the front line a very funny shape.
If the "Race to the Sea" is starting from west of Paris, then surely it will hit the Channel somewhere around Le Havre or maybe Dieppe..
I was assuming that occupying Paris would not change much to the following Race to the Sea which would happen like OTL but just starting from further South.
Plans were made anyway to destroy every bridge on the Seine, which would neutralize part of the strategic advantage of occupying Paris.
The Germans were over extended and exhausted, (as the battle of the Marne would show OTL), especially their right flank.
The French were exhausted as well, and most likely losing Paris would be a blow to their morale but it could also galvanize them when trying to retake it. It would make strategic redeployments more difficult as Paris was already the heart of the railway system. However, it was less so then than we imagine now, because there were
a lot more lines than nowadays.
https://fr.wikipedia.org/wiki/File%3ARoell-1912_Karte_der_Franz%C3%B6sischen_Eisenbahnen.jpg
Now you make me realize that this butterfly might be big enough for changing all that.
Both sides would for sure try to outflank and envelop the other, and at this game it is hard to predict which side would win, but I would favor the French because of the same reasons they won on the Marne + the French more important and available reserves (Gallieni corps which was in reserve in Paris + 6th Army of Maunoury freshly deployed). They could cross the Seine back probably around Cergy and threaten to encircle and trap the 1st German Army in Paris. However, during this war, the French repeatedly missed every chance they had to exploit their victories decisively...
Map of the situation on September 9
As for your hypothesis of a frontline on Dieppe : why not, after all it's very hard to predict how the frontline would move. But if it's stabilized there, that means the French left is beyond the Seine, and there is a dangerous place where they have their left beyond the river, their back on the river, and their right before the river.
As for the Germans, having their frontline so far fragilize them as they need to occupy a huge chunk of France (very populated) with soldiers. Moreover, the whole coastline would be a "frontline" as well, as the threat of a British landing would be there.
The war of movement could likely go on for a while and the trench war we know might as well never exist.
One firther thought. If Paris' main value is as a source of revenue, would there be much point in recapturing it. After changing hands twice (or even once?) t's going to be a valueless pile of rubble.
Well, for the symbol ! Paris remains Paris, even completely destroyed, and any side conquering it would enjoy a huge boost in morale at home and on the front.
I doubt there would be a lot of combat inside the city, as the bridges would be destroyed, the assaults of the French would rather be peripheral than frontal. But if the Germans have to vacate it once, you can be sure they wouldn't leave without destroying and looting everything they can... The Eiffel Tower would be a target of choice, as it had a military value, and TTL could later see Paris being famous for its Art Deco architecture as they rebuilt the city later.
AFAIK, German goals for land would just be that mine near the border. Otherwise just peace and a bit of money. I think if Germany captured Paris, the French would have had to lose quite a bit, and would be fine with a lenient peace. German goals were more chopping down Russia before it got too powerful, so early on in the war, I don't see too onerous demands against France.
Those peace terms are ASB. They would require to change the goals and mindsets of the Kaiser, the Chancellor, the military high commanders, and most industrialists who approved and supported the Septemberprogramm.
The peace terms you imagined sound Wilson-like. No one in Germany thought like that at the time.
EDIT : moved the edition into a new post.
Sorry for the format of the writing, I'm still not mastering the formatting of my answers on this forum when writing on my phone.