Now that I've thought about it, this really could be a cool premise for a TL. Here's a scenario to think about: Gabrielle survives childbirth in 1599. After the annulment arrives (because the Pope isn't in a position to deny the King one at this juncture) Henri marries Gabrielle in say early 1600, scandalizing his nobles and much of Catholic Europe (though Elizabeth I might be privately amused at her former ally's actions). In response, the single Marie de Medici is married off instead to Archduke Matthias, heir presumptive of Emperor Rudolf II. After all, what better way to get back at the French than by allying with the Habsburgs? Plus the only position higher then Queen of France would be Holy Roman Empress. Now depending on who the dowry is payed to (the Emperor or her husband), this could mean an earlier coup against Rudolf by the rest of the dynasty, back financially by the Medici and the papacy. What an earlier reign of Matthias could mean I don't know: if early enough it could change the later course of the Long Turkish war. However, at the least it means that Matthias is very likely to have children and thus denying the ultra-Catholic Archduke Ferdinand the throne.
Now Personally I think a Bohemian revolt was inevitable: the Bohemian Protestants had taken advantage of the Habsburg weakness to get as many concessions as possible, concessions that the dynasty wouldn't allow to remain once they've recovered. But this doesn't mean the thirty years' war would still happen, especially if the Imperialists (being in a better financial position) are able to strike back before the Bohemians offer the throne to Friedrich V. If the Palatinate isn't brought in then the Thirty years' War likely ends at the Bohemian Revolt. Also, with a 1599 POD we could also see a Burgundian Habsburg line develop in Brussels via Albrecht and Infanta Isabel. Hell one could have Allbrecht and Isabel's daughter Anna Maurica survive and marry the son of Matthias, bringing the Netherlands under Vienna's control (assuming Madrid signs off). And this is just what I came up with for the Habsburgs.
On the French side, we have an unstable Kingdom led by a powerful King and an unpopular but skilled consort. Sound familiar anyone? Effectively France would be in the same position as England was in the 1530s. Now this instability could lead Henri and Gabrielle, like Henry and Anne, to take radical steps to fully secure power against the Church and nobility. I can see earlier centralization a la Louis XIV occurring, as well as seizures of various noble fiefs and fortunes. Think the way Henry VIII went after the Duke of Buckingham. Now this could cause further civil war, though more Fronde and less war of religion. Assuming the Royalists win, a civil war would be a good thing: the rebels fiefs, estates and fortunes would be up for grabs and flow directly into the treasury. Depending on how the Church reacts over the marriage and the status of the Royal children born before the marriage, we could see a Gallican Church fully and legally established within France, which was something of a wet dream for the French Kings. Of course this also means France likeluy abdicates any hope of restoring its position in Europe for the time being and any prestigious foreign marriages. Perhaps la Grand Mademoiselle's mother, Marie de Bourrbon, could end up as César's Queen?
One final though. The real chaos in France wouldn't start until Henri IV's death, then we would have the potential of a three-way civil war; between César as the legitimized heir to the throne, the eldest son born to Henri and Gabrielle after their marriage and Henri II de Condé. Or something similar to what happened at Louis X's death in 1314, an Estates-General is called to formally decide the succession.
The mariage of Henri IV and Gabrielle d'Estrées would actually have proven pretty disastrous and complicated for France. Gabrielle was hated by almost everyone but the Green Galant. The Parisian folk hated her because she was an ennemy of the Guise, whom Paris was always very favorable to. The nobility hated her because she was spending a lot of money. Gabrielle herself was the object of numerous Pamphlets. Marguerite de Valoirs, Henri's first wife, didn't want Gabrielle to replace her because she felt she was below her rank. And Pope Clement VIII probably wouldn't have been happy because he was already planning the marriage of Henri IV to his niece, Maria de Medici. In other words, you would have a very scandalous marriage and Gabrielle would have been a very unpopular queen... And unpopular French queens don't do well: see what happened to Marie Antoinette.
I'm not saying Gabrielle didn't have qualities but the fact remains that no one but Henri IV wanted her to be queen of France. The move would probably have undermined Henri IV's actions to pacify his kingdom and restore its power. The last Religious War had ended in 1594 and Gabrielle died OTL in 1599... I'm not saying we would still get something as violent as a Religious War but fact is that we're left with a Kingdom were tensions still run pretty high and with parts of the nobility that could have risen up against their King for such a move.
Aside from that, there is also the question of France's economical situation at the time. The main reason Henri IV married Maria de Medici was actually because she came with a very advantageous dowry. Hell, Maria even ended with the nickname of La Grosse Banquière (The Fat Bankress) at court. Gabrielle wouldn't bring such a dowry. Now, saying Maria de Medici's dowry solved France's financial problems is probably pushing it but it did help a bit but there might be a few troubles coming from marrying Gabrielle d'Estrées in that regard.
So in a way Gabrielle would be France's Wallis Simpson. However, the Marie Antoinette comparison doesn't really fit sense she didn't start out hated by the entire country, as your claiming for Gabrielle. Marie de' Medici was unpopular, as was Catherine de' Medici but they were fairly successful. This isn't the 1700s. As to the unpopularity, Tte moderate nobility are likely to reconcile with the King at some point, though nothing could be done about the Parisians. The rest of your points are addressed above.