Anne is fine as name for a daughter of Francis and Mary (maybe Catherine, Antoinette or Elisabeth would be more appropriated but Anna d’Este, Duchess of Guisa was Francis’ second cousin and Mary’s aunt-by-marriage)
True but you would think they could use Claude...
Pretty unlikely... Anne is the best compromise name as would honor both branches of the family (and I can not see Mary being happy to name her daughter Catherine)
I had not thought to Anna d'Este (who was Francis' first cousin once removed, not his second cousin) but she was only a pretty vague relative. Francis and Mary naming their daughter after her seems a little farfetched. Except of course if she is chosen as godmother which is definitely possible.
Claude would honor both sides two as it is the name of both Francis' paternal grandmother and Mary's maternal grandfather.
As for Catherine, that's still a possibility as Francis is likely to be the one to decide.
Pity who Catherine here is an “Italian merchantess” without ANY RIGHT OR CLAIM to the regency for her grandson. Her only power was the (limited) influence who she had on her late son. In OTL Catherine got lucky with Henry II and Francis II’s premature death who left her in charge as regent for the still underage Charles IX but here she. will be powerless as nobody would give her any power in the regency for her grandson.
Mary will be sole regent as Catherine was in OTL
First of all, Catherine was NOT regent for Charles IX in OTL. She compated for the position against Antoine de Bourbon. The Parliament of Paris choose Catherine but REFUSED her the title of regent and instead gave her the innovative title of "gouvernante".
And saying "Catherine got lucky with Henry II and Francis II’s premature death" makes no sense, even without taking into account the pain she suffered from losing her husband and her son. These deaths meant Catherine had to do the job herself in order to protect the children she still had and had to do what she could to rule the country with very weak authority, many people despising her ( the “Italian merchantess” stuff was quite an issue).
Catherine's position was much more comfortable before Francis' death than after. Before, she was the listened unofficial advisor of the undisputable king. After, she was the despised and soon hated "gouvernante" (not even regent) in precarious situation. It is not fortuity that she put Charles in charge as soon as she could, even moving forward one year the time of his majority.
In the scenario that Francis dies young having an underage son by Mary, who will be regent of France is an open question. I know you believe the mother of an underage king of France is automatically regent but it is not true.
In OTL, despite being Charles IX's mother, Catherine needed to be appointed by the Parliament of Paris against Antoine. And she was not even made formally regent.
In the case of Mary, everyone would know she would end up as her uncles' puppet. And, the Guises, despite being popular in lower classes, were hated in nobility. They were seen as ambitious foreigners (Lorraine was not part of France at this time) having usurped positions that should go to French families like the Bourbons and the Montmorency. Obviously, all opponents to the Guises would dislike the idea of Mary becoming regent.
Of course, Mary has her chances but so does Catherine and so does Antoine.
Except Scotland was also rapidly becoming uncontrollably "heretic" in Mary's eyes and also began to rule in a way that would NOT fit Mary's interests (pro-France, Roman Catholic)...
This is the point. Mary would not let Scottish Protestant rebels do what they want.
The other option is she sends in foreign troops, which likely winds up driving the opposing Scottish forces into an alliance with the English and could mean Elizabeth I of England is forced to support Hamilton or one of Mary’s other Protestant rivals.
I think this is by far the most likely scenario.
NO, She DO NOT NEED to do personally anything about Scotland. She has Regents and the French army for keeping control of Scotland
If they succeed.
both her elder uncles were good military men
The cardinal of Lorraine was good military man? Are you sure you are not confusing with the third uncle, the duke of Aumale?