If Louis XIV had been born in 1619, he would have been a different guy. For one, he would have become King at the age of 24 so he would be a grown up man and there would be no need for a regency. Second thing is that Anne of Austria's role in his education would probably have been reduced a bit because of Louis XIII and Cardinal Richelieu (I don't see Richelieu's rise being butterflied here). Anne wouldn't necessarilly be out of the picture though: she and Louis XIII were at their closest in their relationship in 1619 thanks to the fact the Duke of Luynes and his wife (the future Duchess of Chevreuse) were close friends to both.
The goal of Louis XIII and Richelieu had been to increase royal power, and improve the governance of the realm. OTL Louis XIV followed in on their footsteps, so it's likely TTL Louis XIV would do the same. Where they will probably differ are on the methods because they won't have the same personnality. Depending on how events play out, we might not see a revocation of the Edict of Nantes for example, though TTL Louis XIV might still look up at ways to reduce the power of the Protestants. TTL Louis XIV might also not be as warmonging and glory seeking as OTL Sun King, though he would definitely want to have France a major power if not THE major power in Europe.
In terms of immediate change for France, the first big one might be with the Fronde. The Fronde has a chance of not happening because the transition will be a bit smoother: the King here will be an adult with a few political years behind him, possibly military experience (Thirty Years' War), possibly married and with a secure succession, and it's likely we would have another King-Cardinal duo in charge (Mazarin being Richelieu's designated successor). Even if the Fronde isn't butterflied away, it probably wouldn't be as huge a nuisance as it was OTL or at least events could play out differently. The second immediate change might be the Franco-Spanish war. It's dependant on the Fronde of course, but the conflict will be affected by having an adult Louis XIV succeeding his father. The French could potentially do better against Spain if they have no internal troubles.
Regarding the potential bride for TTL Louis XIV, I would personnally be looking for one in Italy or Catholic Germany. Among the possibilities, there is notably Louise-Christine of Savoy (1629-1692): she's the daughter of Louis XIII's sister, Christine, and Duke Victor Amadeus I of Savoy. The situation in Savoy at the time might lead to such a match if Duchess Christine want to strengthen the alliance with France or feels the need to have the support of the King. The main problem though is the ten year gap between the spouses in this scenario.
Most likely only as a last resort if a more suitable match can't be found in the other courts of Europe. Doubt it would come to this.
TTL Louis XIV marrying a Hapsburg to secure peace is still among the possibilities though: it wouldn't be the first match resulting of this. Trouble might be on finding a suitable candidate: I don't know if Emperor Ferdinand III has an available female relative at the time and King Philip IV of Spain's daughters are too young here (Maria Theresa, the wife of OTL Louis XIV was born in 1638 and she is Philippe IV's eldest surviving daughter).
Actually, Henri IV's original plan was to have Louis XIII marry Nicole of Lorraine, daughter of Duke Henri II of Lorraine. This would possibly have allowed for a peaceful annexation of Lorraine by France. Given that Henri IV died before that though, the plan never came to fruition and even then, the succession of Nicole in Lorraine was hugely contested (to the point she was eventually overthrown by her own husband and cousin, Charles IV of Lorraine).
A match with Lorraine however here is highly unlikely. For one, there are no real potential brides for TTL Louis XIV among the Ducal family. Second, and probably the biggest problem, Duke Charles IV of Lorraine wasn't on the best of terms with Louis XIII: as a matter of fact, he often supported Gaston d'Orléans, Louis XIII's always conspiring younger brother, to the point Gaston endedup marrying one of Charles IV's daughters, Marguerite.
Her being eight years younger than TTL Louis XIV might be a problem in the grand scheme of things. Plus, as reliable as the United Provinces are as an ally for France, the match might not be considered that prestigious. And there is the problem of her having to convert to Catholicism: not necessarilly a huge problem, but still a problem.
The Stuarts being cousins wouldn't really stop the marriage: OTL Louis XIV married his double first cousin Maria Theresa of Spain and his brother Philippe's first wife was also a first cousin and a Stuart to boot (Henriette Anne of England). Plus, first cousin marriages were pretty common.
That being said, I agree that most of the Stuart princess aren't the right age to marry TTL Louis XIV.
I'd personnally say pretty unlikely. Assuming Gustav Adolph does end up having a son, Christina is still second-in-line. I think both her father and the Swedish nobility would favor a Protestant match for her.
There are also troubles on the French side to take into account: Sweden is an ally and a powerful Kingdom, but I still think Louis XIII would look to secure a Catholic match first and foremost.
Okay interesting, so a Mazarin Louis 14 duo working together certainly very interesting. What might relation between father and son be like?
As for marriage options which other Italian or German brides are their available?
As for Gustav II if he has a son whom might this kid marry?