Gregor had an older brother who died before the start of Shards of Honor.
If he were Prince Serg's oldest child, standard Barrayaran naming conventions would imply that he should be named "Ezar". He could be a second son — Bujold hasn't written anything about Princess Kareen's family, so "Gregor" could easily be her father's middle name...
- Possible, but I think it would have been mentioned. As a parallel to Aral's second-son-ship, at the least. Perhaps the naming after grandfathers is the custom among the nobility but not royalty, or custom in rural districts like the Dendarii mountains, or some such. It's unlikely to be a universal custom for the entire planet, at least.
- Besides, according to the same custom, everyone has two names, not one. For all we know, Gregor is actually Ezar Gregor, or Ezar was actually Gregor Ezar, but this is never mentioned - after all, somehow no-one managed to notice the correlation between Admiral Miles Naismith and Miles Naismith Vorkosigan (just how bad is information access between planets? in the modern internet, finding out Naismith's identity would be as simple as googling "Miles Naismith" and looking for portraits - he is a pretty prominent figure in Barrayaran politics since birth due to inheritance).
Ivan's going to marry a Galactic
He's about due another stint of embassy duty, and he's pretty much run through all of his prospects at home. Plus now that both Gregor and Miles have produced heirs, he's out of the crossfire, and
being physically normal he's under less pressure to be conventional about such things.
- The Amazon description of the next book heavily supports this, especially since it's about Ivan, and features him rescuing a female Galactic.
- Confirmed. Tej is half-Jacksonian, half-Cetagandan, and is a granddaughter of the last Cetagandan military governor of Barrayar to boot.
Part of the reason that Simon went along with the plan to break into the vault beneath ImpSec HQ was because he was hoping they'd damage it somehow in the process
He'd hated that building for his entire career, and had been trying on and off to get the funding to replace it for thirty years before his efforts were ended by his retirement. If there was a chance he could finally get rid of that eyesore once and for all, then he was going to take it.
Kou's Wedding budget woes were solved by Lady Alys insisting on holding the weddings at Vorkosigan House, Vorbretten House, or the Imperial Residence
Gregor is likely to be a wedding guest after all, so the location has to be secure. And Lady Alys is the Emperor's social secretary and still owes a debt to Drou and Kou
The Imperial family is part Cetagandan
When the vault is breached, it's mentioned that the contents of the letters contain information that is to be marked confidential and could badly shake things up in the current system. My theory is that it includes proof that the Cetagandans tinkered with the imperial bloodline, either through interbreeding or more subtle methods. It would help to explain Gregor's hyper-competence (and possibly Serg's bloodthirsty insanity).
The Dendarii Mercenaries are going to found an empire
In Real Life
when a mercenary organization gets tired of fighting other people's wars they often simply find themselves a nice cushy kingdom to take over and declare themselves kings. Why should not the Dendarii do the same? They probably have more experience and firepower then many small nations, and many would rather be ruled by Queen Eli Quinn then by King Whoever They Had Before. Besides many employers of mercenaries tend to default on their contracts. And what do mercenaries do when that happens? They foreclose! Forcibly.
Either or both of the Kosti brothers have or will become Vorkosigan Armsman
Because Miles just inspires that sort of loyalty in people.
- Ma Kosti will be come a Vorkosigan Arms(wo)man, in order to keep various high-ranking relatives from stealing her away from Miles. Besides, "Vorkosigan does it backwards" - instead of having armsmen who are "cooks" like Count Vorloupulous, Count Vorkosigan will have a cook who's an "armsman."