OK you are WAY overestimating the power of the Parlement de Paris. It may have saw itself as the Premiere Parlement but it can be overruled.
Beg your pardon, but I think you don't have a clear view on the situation in Paris duing the Wars of Religion. The Parliment had a potential power quite important, and if it was shut down during periods of strong royal power, it was a quite powerful institution in periods such as troubled regency.
And it was not about "seeing itself as the First Parliment". It was seen such by population, others Parliments, and judicial elite.
Before Henri IV created ex nihiilo a rival Parliment in Tours, it had a decisive influential power among the population of northern France that saw it as its representative.
By exemple, the Paris Parliment was trusted to secure and rule Paris in the absence of the king at times during the XVII century.
If generally obedient, while reluctantly, it's certain that if the sucession goes wrong at some point and seeing how much Paris is of a Guise stronghold, the Parliment could gain power (whilst under Guise/Ligue influence as OTL when it refused to consider Henri IV as king and choose Charles de Bourbon instead).
So yes, I stand my point, a troubled situation as a regency for Charles IX's son could lead to a Catholic (not religiously, but politically Catholic, you got the point) Parliment to challenge Catherine de Medicis regency and could lead to prefer a novice. Hell, for Guise and Catholics parlementaries, a catholic novice could be even better as it would gave more room for expanding their power rather than a skilled and experienced "Politique".
Never said it was the most realistic, though: as you said the OP is vague. But in the worst of case for Valois (apart, of course, being extinct) it's a real danger. I don't think that a generalisation of instituions like the "Provinces de l'Union" as they existed in Languedoc (while not widely accepted there) is totally unrealistic.
A better situation, with Valois willing to make concession with parlementarians in order to prevent a too great Guise influence would be to agree more or less partially with the point of view of Parliment regarding its institutional role (aka a permanent royal council and the "constituionalisation" of its rights that were largelly informals).
I'm not sure I get this part. I mean yes Henri of Navarre had vast estates in France, was technically Primer Prince of the Blood and was the leader of the Huguenots but he wouldn't be the Heir to France.
-Bourbon-ALbret were powerful princes, with large holdings that were basically autonomous.
- The policy of french kings was to limit the power of the great houses within their kingdom.
- The fact Henri of Navarre didn't join forces with "politiques" like it did OTL and remained the leader of an opposite faction from the royal one makes him a threat.
- Valois could want to both "sacrifice" him to Catholics and to get rid of an important prince, like they did by confiscating Charles de Bourbon Estates : no matter the pretext, they'll find one.
He would still be powerful but part of the Ultra-Catholic response wouldn't be as strong without the threat of a Protestant King of France.
It will be the same. Without the threat that representated the cheif of the Protestant faction becoming the legitime king of France, they'll have more room to get rid of him in a legal way, and to make a (temporary) alliance with Politiques on the base seen up there.
In common opinion of historians, they didn't.
Again, Catholic were mainly Ligue and other little factions with a clear program : get rid of Protestant politically and religiously, have the king of France following a pro-Hapsburg and pro-Catholic foreign policy, restrain the royal power at the benefit of great house and local (including parlementary) ones.
Not one of this was reached. I'll repet myself but Wars of Religion became soon less and less about theology and strict religious policies but about the political struggle of great houses and local powers against royal one, an usual conflict in France up to Louis XIV.
Saying it was Catholics vs. Protestants, critically representating them as solid and united faction is wrong.
Again, the clear victor of the Wars of Religion was the "politique" side.
Think about it. Did France go Protestant? No. Did France have a Protestant King?
Wars of Religion were more about a power struggle to decide who will be the most influential faction in the court : Guise, Montmorency, Bourbon, Valois themselves...
Basically (really basically) the Catholics were the faction of Great Houses and populars classes, Protestants of the gentry and some urban bourgeoisie, Politiques balancing from each, depending on their focus.
I'm not saying religion didn't played a great role (it's the Wars of Religion for some reason after all) but the political part of these had others roots.
The Protestant faction didn't had the conversion of the kingdom as objective to the war (Admittedly, it was an ideal objective, but certainly not immediate.). It was maybe what Catholic then Ligue said, but it was total BS.
Their motivations were more to keep Protestantism dominant where it was present (by exemple, refusing the reconstruction of Catholic clergy and buildings where they were destroyed or deserted, something that was refused in the Edict of Nantes), to create a stable Protestant faction within the court to protect their interests. Finally to create "a state within the state".
Did the rights granted to the Huguenots last?
Again, the prescripton of Edicts of Beaulieu and Nantes weren't about giving "rights" to Protestants as individuals.
Not only they didn't fulfilled all the objectives of the Protestants.
By exemple, it doesn't allow freedom of religion, but freedom of conscience.
Basically, you're allowed to belive whatever you want, but you have religious rights only in precise places (in said places, contrary to Protestant wishes, Catholic religion was authorized to be re-established) and forbidden elsewhere.
Finally it limited the religious and political autonomy of Protestants in sort of "reservations" where they had to share power.
The conditions themselves were made in order to satisfy each faction and to make them recognize Henri IV and Politiques rise.
and were eventually banished from France altogether.
No. Not even by exageration.
See, some Protestant indeed exiled themselves while it was forbidden by the royal edict. They were supposed to stand in France, and exile was outlawed.
To resume : banishment was the contrary of what Louis XIV intended.
See the article 12 of the Edict of Fontaineblau for more precisions.
And the departure of french Protestant was finally limitated : at worst 100 000 (essentially in North and East parts) aka 1/2 of the exiled Protestant during the Wars or Religion.
The crushing majority of Protestants remained in France, mostly as crypto-protestants or as rebels.
Finally, even before the death of Louis XIV, the persecutions began to slow and it's realist to say (according contemporary sources at least) that the Protestant cult began to be restaured semi-clandestinly (while it stopped almost totally or turned to eschatological turmoil) in the 1730's
Nevertheless, as the Wars of Religion ended with Edict of Nantes, in 1589, talking about Louis XIV policies one century later to talk about the results of the former doesn't make great sense : it would be like saying France won war of 1871 because it eventually took back Alsace-Moselle in 1945.
As for the future of France, I think it depends on what type of King Charles' son is. If he's a Louis XIV, Francois I or Henri II, then France's future looks bright. But if he's like his father or uncles, or to a lesser extent Louis XVI, then France is in trouble.
A Louis XIV with a country divided in rival factions having in common the refusal of a strong royalty isn't going to go that far.
That said, Valois were particularly despised in french historiography up to nowadays, partially thanks to Bourbon propaganda, but they were fairly able kings : Charles IX, Henri III by exemple did a great part of the job that allowed Henri IV to rule efficiently.
They had hard times tough, and Charles IX son is going to, as well.