Lavrentia Rising

For the first question, yes, that much is true. Not all planets are going to get concerned with getting attacked. Some are pretty much just bystanders in this big galaxy, and only have to care about lending troops to the war effort of who's ever in charge at the time. Some planets, despite being part of major powers, mostly keep to themselves, especially those under the Arkadi. Some don't even make sense to defend. Yet, there are planets which of course have to be assaulted in order to take control.

For the second one, I haven't really gotten into the economy that much. When I was fleshing out the world originally, it was just for the sake of writing, but as it grew into an actual project for me, I haven't really touched on the economic things and the goods that they sell, but yes, in my mind there is a case in which there are Lavrentine goods being sold in Arkad, and as a competitor to the former hegemon, it is able to spread its own products to increase its economic strength and influence. After all, most of Lavrentia's conquests succeeded because they conquered diplomatically and economically. Before we get into the culture thing, I'll just clarify that Lavrentia is a single government/nation comprising of several former backwater planets, which were highly developed under Lavrentine rule. But yes, it is a hegemon in its own right, influence nearby regions as well as the unintegrated planets within the region itself. Also, nothing really of note for the cultures, but for the sake of worldbuilding here's what I've come up with so far:

For the culture thing, the region that comprises Lavrentia actually used to have similar cultures, albeit slightly different languages as they evolved separately. Dzhalus, which is at the core of Lavrentia, is known for having its language and culture heavily inspired by the Medinians (read up on the history part heh), as well as the Arkadi. The rest of Lavrentia never truly managed to completely adopt Imperial culture. There have been minor effects, but none of them as major as in Dzhalus. This is because this region was left to fend for itself without much technology. In the case of Dzhalus, it was an uninhabited, recently-terraformed planet. The settlers sent to this new region were mostly racially homogenous, and when their ship's hull had been turned into the temporary living quarters, the major Dominions war caused communications to be cut with Dzhalus. They couldn't survive on technology, so they began hunting. Two centuries later, Dzhalus forgot the times when it was part of a wider humanity, and its culture developed on its own from the old beliefs of their ancestors. They focused on building their own societies and tribes, whilst adopting ancient methods of warfare. This was further accentuated when a few of the original Lavrentine planets' settlers managed to rebuild their ship around the nuclear reactor before one generation passed. They flew all around the region for centuries, watching other cultures develop as their own did, as well as influencing them. Sometimes, they even warred with the other tribes, and when the ship finally ran out of fuel, the interactions between planets stopped. The original culture for what comprises modern-day Lavrentia was heavily martial and superstitious in nature. For example, they had elder priests and priestesses who gave omens before battle, their planets were separated into distinct sectors which usually warred with each other, they were interested in pottery and kept the deceased ashes in them, they had a religion based off nature-worship integrated with the ancient religions of their ancestors, etc.

When the Medinians under Emperor Dzhalus came and influenced the backwater planet, they also imparted on them a culture of imperial might. Think European culture from the 1500s-1800s, but in the future. Limestone was gathered from the lakes, and steel forged from the mass reserves of iron in the undiscovered mines. They built massive structures. Most notable of these is the Wall of Gran Medinia, south of Dzhalipol, and the capital city of what would become the planet of Dzhalus, as well as the Gran Ekklesi (the Medinian term for "Great Church" at the centre), and the Hall of Dzhalus, which became the palace from which Dzhalus' governors could rule the planet directly. The other Lavrentine cultures were inspired by the other Imperial nations.

As for Arkad, their culture is pretty much based off republican revolutions. In the early years, the Dominions didn't properly handle Arkad despite it being a well-developed planet. Many of its citizens were independent, and had many republican philosophers rising. It revolted, but the Dominions navy quelled this rebellion, but only crushed them to the extent that the Arkadi wanted revenge. Arkad also has a rather martial culture with soft expansionist policies. It wanted to spread the revolution and be the ones to control the peace, as they believed it was some sort of divine right. While nobody believes it is a divine right anymore, most Arkadi nowadays believe that they were to most capable of doing so because they were better and more effective. This unwittingly created their secret lust for galactic economic domination due to the nature of Terrans (who dominated the upper class) wishing glory for their nation and their kind to the point that Terran and Arkadi culture had blended together. When Terra became independent, many native Terrans flocked back and established a republic. Terra itself, despite having exhausted its resources centuries ago, had a government filled with power- and money-hungry politicians that sought to dominate the galaxy economically, with the help of the Arkadi military. However, Arkad itself is an economic power in its own right, specially radioactive fuels and weaponry to other nations.

The biggest defining feature of what can be called modern Arkadi culture, however, is Neodemocratism. Think hipsters and hippies, but their culture is now a religion and has spread galaxy-wide. Basically, drugs used as an escape from suffering, extensive pleasure-seeking that has managed to destroy any meaning in life, and social egalitarianism (though this last one was unable to be effectively managed to the point that it became a theory and never put into practice again).