Last winter I read a biography of her by Leonie Frieda. She touches on these three topics though not in the most detail (especially in the case of the first two). She does make it sound very much like she used her flying ladies as spies and at the very least to garner favor/some control over powerful men at court. Regarding the dwarves they just seemed like an odd curiosity and Catherine actually had close relationships with them, she had favorites of the bunch. Apologies I can't recall more. I recommend the book for further reading as it was well written.

Francis kept his court of women and the main thing I took away was this was essentially where Catherine was able to learn (from the King and the other women at court) how the whole process operated and how she could use her charm and intelligence to manipulate the system to her own ends. At first much of this was done simply to ensure she would be a strong queen for her husband and children when he ascended the throne.

Important note, both Francis and Henri had lovers who they adored and kept pretty openly. These women had huge power over the kings and I think this was a lesson Catherine took to heart when applying her own court of women. She also saw how quickly Francis' lover fell out of power when he died and I think this drove some of her quick strong action to ensure the same couldn't happen to her when Henri died. At the time his lover had her own supporters at court.

I think regarding her children it was rather a few overbearing mother issues as well as the difficulty of her times.

She strived to install her family into the monarchy and wanted nothing more than a permanent lasting dynasty of her husband's heirs. With the early death of her husband she needed to take a very active role in maintaining power while they came of age, which in my opinion caused her to develop a desire to remain the all powerful ruler behind the face of her children as king. As her children came of age this caused instances of internal strife as they tested their own power. Though I will note, especially earlier on in her life, her sons tended to not take such an active interest and let her run much of the show.

When she lost her first son she would have been very aware of the precarious position her dynasty was in just upping the anxiety of hers to protect her remaining sons and make sure they produced heirs. She was well aware how sickly her spawn was as it was the Medici trait.

Henri also seemed to be her favorite due to his personality and devotion to her from a young age. I think she also saw more of her late husband in him at least while he was a boy and this cemented it. Because of this she harbored dreams of him being a king which I think were the basis for many of the family issues.

I think as Charles lingered on the throne she grew even more anxious that this would mean less chance for Henri to ascend. And Charles wasn't producing heirs while his health deteriorated. Charles and Henri also had a hearty sibling rivalry and she sided with her chosen, causing tension. I think Charles was the first to really start to take some authority from her as well but this could use a refresher from my source.

Regarding Margaret I think it comes down to a few issues. First of all she was rebellious and helped the younger son Hercule in his plans to make a name for himself with the Protestant cause. In doing so they were both placing her hard work at dynasty making in jeapordy, threatening her son's life, and also continuing to pick a scab that Catherine had worked most of her life to soothe (the religious divide and civil wars through her reign).

Hercule was essentially shit disturbing trying to drag France into open war with Spain's holdings in the Netherlands for a protestant cause. There was basically everything for Catherine to lose in this scenario and while Hercule was being a bit of a selfish prick, Margaret seemed to be going along with it out of spite towards her family. And as this was unfolding when Henri was on the throne, you better believe this soured Catherine on these two real quickly.

I could also see Cathirine being bitter that while her blood would live on through Margaret, it also meant that Henri of Navarre and the rival bourbon line would ultimately win the throne. She loathed this outcome as it meant her beloved husband's line was dying.

I don't know. I'm sure more knowledgeable posters can pump this full of holes but that was some of the broad strokes bullet points in Catherine's family dysfunction.

Tl:dr A hunger to secure dynasty with a very sickly brood and a backdrop of religious wars crippling the political system in France caused the over bearing mother tension and the screwed up relations between Catherine and her kids. Those she viewed as useful to securing her husband Henri's dynasty were overally ok and those working against it she soured on.

I'm very interested in the psychology behind why so many Renaissance ladies, in particular, kept dwarves.

Unfortunately I don't know enough about Catherine de Medici specifically to address the rest of your questions (I wish I did, believe me; she's a fascinating historical figure). But I can talk a little about the broader context here.

The later 16th century is still part of the era when queens maintained their own households and courts. The purpose was to promote the queen as a power to some extent in her own right, but even more so as a proxy show of the king's authority. Additionally, flashy exotica, such as animal menageries, had long played a role in the display of royal power (on the kingly/princely level).

Renaissance-era European exploration/expansion escalated this latter tendency to new heights. Courtly retinues needed to be filled out, numerically, by servants and slaves: some of them served practical day-to-day functions, but others were more for show or had some sort of practical use kind of dumped on them. The 14th-16th century courts in Spain and Austria, in particular, leaned on sub-Saharan African slaves for display. For example, chronicle accounts pay as much attention to what the black slaves are wearing as to the garments of the queen: it was about making a statement of presence. Dwarves, a different type of exotica in the late medieval/early modern mind, served a similar function.

I'm sorry I lost track of your question until now! I know someone who has written on Catherine de Medici in the past from the religious/power angle--I'll ask her if she has more insight into your specific questions (which are now also my questions), but I kind of doubt it.