Her Imperial Majesty Emilie II of House Dimka-Tseng, By the Grace of God Mistress Of The Empire, Custodian of Earth and Defender of the Faiths, and the rest of the upper echelons of House Dimka-Tseng receive the same level of tabloid attention as the current British Royal Family, pretty much across all the Civs.
Her younger brothers Joshua and Kamil both have proud military records (not that either of them is generally believed to have seen combat) and since completing their service have been generating a catalogue of highly entertaining, column-selling, but ultimately unimportant scandal. Her Majesty has yet to wed, but at 40 she still has plenty of time ahead of her and only the most decidedly conservative news outlets have bothered editorialising about it.
The Imperial Palace covers a fair portion of what was once Paris, in western Yrasie, and public functions and appearances tend to take place in the Eiffel Dome.
Any imperial citizen can get a travel visa for earth, much of which is a massively overpriced tourist trap, particularly for the nobility where there are a variety of places one just has to be seen going. “If one hasn't spent 200 guineas on a plate of greasy MSG laden noodles from the 'faithfully restored' 20th century Camden Market, or played roulette in the 'Royal Hive' (a really racist Many themed casino in Las Vegas) one simply hasn't lived darling.”
Following the Mecca incident citizens from other civilisations cannot apply for visas for Earth outside of diplomatic functions. The Zebulons “cultural outreach centre” is the first change of this policy and has caused quite a stir (and only passed due to being partly ‘diplomatic’). In practice nobles can get visas issued for ‘their guests’ with sufficient bribery, but that noble is going to be personally responsible for whomever they have brought: while the scandal of taking the latest fed rockstar on the London tour can enhance one's reputation if it comes out afterwards, if said rockstar makes a scene and gets arrested it will go very badly.
First child is the default inheritance of a title, but it's common for people of any class to make their wills favour children who have brought benefit to the family as a whole, and this applies as much to titles as cash or land. Gender is not a default official factor in the process.
Same sex relationships are normal and entirely accepted (as are people who don't carry their birth-assigned gender, incidentally), and biosciences mean that there are plenty of options for production of heirs.
As would get dispatched to Redulutto it would be unusual for the actual baron or baroness to risk themselves on the planet, “children” trying to win approval to be named heir are much more common, speech marks because due to advances in medical science, the existing title holder can live a long time. (See Prince Charles and the current queen). Similarly, trusted siblings or loyal knights from families that owe their allegiance are the most likely to be sent to represent a house that gains imperial charter to exploit the planet.
Speaking on the rest of the family: given inheritance can and does change depending of the will of the title holder on death it is traditional for the various children to be addressed using the generic lord/lady title and treated with the same respect as their title holding parents (if for no other reason because you never know who is going to end up in charge one day). Once the inheritance happens it is customary for the other siblings to be given knighthoods, whether these are honorary or actually come with management of a district depend on the competency, inclinations of the sibling, and how much they are liked by the one that did inherit.
Common career choices for nobles who are not in line of succession:
The rank structures most Houses use will be instantly recognisable to anyone who knows a little about the modern British Army.
It is unlawful for anyone of common rank to hold a military rank above Lieutenant. However, this is one of the few reliable ways outside of suddenly developing NX powers to move up in the world. Upon selection for promotion to Lieutenant a soldier will attend the same military academy as young nobles preparing for their service. While the nobles there will be learning to be officers, they will already be gentlemen or 'young ladies at arms', the commoners are there to learn both, so that if -as is rather expected of them- they perform well enough as officers to earn a knighthood, they are prepared.
Military discipline will vary from House to House, and even commander to commander. While obviously there will be times when a Meeting Without Coffee just won't cut it, abuse of the lower ranks, even in uniform, is more than just a faux-pas, and nobody wants a neighbouring House to launch a police action on you because Lieutenant the Lord Okembe had Pvt. Tseng flogged over insufficiently shiny boots again.