Probably the biggest thing for a GM using the Third Imperium to get a grip on is figuring out how the Imperium works. Frankly there is a lot of ways it can happen. From where it doesn't do much at all to a really dark empire that is a reflection of the Star Wars Empire.
This is my take on it.
The Third Imperium was established to allow business to make money on a huge scale. Not just subsectorwide or sectorwide but stretching across multiple domains (4 sectors in a 2 by 2 grid). The watchword of the Imperium is stability. The early Emperors achieved this in two ways. First by allowing the member worlds near autonomy in their internal affairs. Second by creating a nobility that is loyal to the Imperium as an ideal worthy of defending with their lives. These two factor combined in creating a stable empire that spanned dozens of sectors. This allowed trillions of credit to be made in good and services. The idealism of the nobility was reinforced by the prosperity they enjoyed.
My Imperium is about commerce. Creating it, and protecting it. It not about a short term view either. Many nobles and corporate execs will forego billions now for trillions later.
What it means for your game that the Imperium is rarely involved in individual worlds. They operate the official local starport and will patrol out from the starport to the 100 diameter limit. The rest of the system is the responsibility of the local world which usually has only a few outposts. Starport are freeports. Their budgets are partly subsidized by fees for fuel and berthing and the rest covered by the subsector budget. Their value comes when they are used by the various megacorporation own by the emperor and the nobility.
Starports have only a handful of restrictions on what you may carry or trade. Slavery and nuclear weapons are two big no-no. And if you try to pull a gun in the starport bar likely a few minutes later a janitorial robot will be cleaning your brains off the wall after an Imperial Marine has blown your head away.
Just outside Stargate the local world will have established their customs area. Once there you and your good are at the mercy of local law and custom.
The Imperium collect a tax based on the Gross Domestic Product. This provides the vast majority of funds used in the subsector, sector, domain, and imperial budgets. Subsectors are primarily responsible for maintaining and administering starports. Sectors are primarily responsible for sector wide defense and responsible for maintaining any military bases including scout bases.
Domains have recently reemerged. They were originally founded as a planning and logistical center for the initial expansion of the Imperium. Later Emperors made their role mostly ceremonial and relegated their task to the sector or imperial level. Emperor Strephon has reactivated the domains restoring them to their original roles. To this end the Domains now maintain and administers the central depots. Depots are entire systems devoted to manufacturing, maintenance, and repair of the Imperial forces.
At the Imperial Level are the vast central bureaucracies. Most of their task involves the collection and processing of data to be relayed to the Emperor, and the drafting of technical specifications for use at the Domain, Sector and Sub-sector. There is very little of what we consider regulations in the modern sense. The Imperium is simply too vast and too diverse for any central authority to issue specific instructions.
For example the Imperial Ministry of Justices maintains a variety of databases, publishes several dozen specialized journals on different areas of Imperial Law, high tech level labs, and forensic research establishments. The result of all of this is shared (slowly) among the various law enforcement units at the sector, subsector, and planetary level.
The Imperial Army, Navy, and Scouts operate much like the central bureaucracies. Leaving Operational control at the Sector level. The central staff used to be involved in planning but Emperor Strephon has pushed that down to the Domain level. What planning remain is about logistics.
What this means that the Imperial Nobility and Officials have access to 4 star support for just about anything they attempt. Anything new or novel they had to overcome get fed back up the chain to be disseminated to the rest of the Imperium. The process is slow but the results is second to none.
So how is policy is made? That where the nobles come in. The Emperor in theory could give a direct order to a Sector Admiral. There is no chain of command like in the US where the President order the Secretary of Defense who then orders the Joint Chief of Staff who then order the Theatre Commander and so on. The system is setup so that the Emperor will take the advice of his staff and then issue the order directly to the Sector head.
However the speed of communication is such that this process can take up to two years from when new was sent, and the order was given. To overcome this the Imperial Nobility has been created. Each Imperial Noble knows they are the personal representative of the Emperor and are expected to act as such in daily business and in times of crisis.
The Nobles operate outside of the normal chain of command. They may have some specific position but they are expected to look beyond that and deal with any problems that come up.
For example Sir Albert has a nice job as the Starport Administrator at Ashdown. He notices that insurance rates keep going up on starships registered at his port. He looks into it and finds out that it because there has been a rash of hijackings and disappearances in the last two years. A query is made to Sector Command which will take a couple of weeks. Meanwhile at a few social gatherings he talks with other nobles and finds out there is something going on. Sector Command comes back saying there is "No Problem."
Sir Albert then requests an audience with the Subsector Duke. The Duke looks over his data and his reports and agrees that more is going on. The Duke then sends a message to Sector Naval Commander insisting that they look into the matter. Also sends a message to Sector Justice that a law enforcement taskforce needs to be convened to investigate why Sector Command hasn't picked up on this.
Within a few months it was found a that a group of pirates have been stealing and hijacking starships. They had a man on the Sector command staff that altered the data so that the losses wouldn't raise any red flags. The Duke congratulates Sir Albert and recommends to the Emperor that he be raised to a Baron.
Nobles are the Emperor's enforcers. Their power comes form the fact that to cross a Noble is to cross the Emperor himself. Now nobles can and do come into conflict over policy. In which case it a matter of being Machiavellian as to which side wins out. Whichever one does that will be Imperial Policy.
The different ranks of nobility comes both a sign of social status but also of level of expected responsibility. A Duke is expected to watch over a subsector. A Count a group of worlds. A Marquis an important world (high-tech, high population). A Baron a world. Knights have more limited areas. Often assuming certain positions carries with it a specific noble title. For example Sector Admirals are often given a title of Duke. This is so that they can deal with the other dukes of the sector on a equal social level. These "rank" title are lost when the person leaves the position. If the person had an honorable service then the Emperor will generally reward the next rank lower on a permanent basis.
Emperor is the owner of many worlds, corporations and properties as many nobles. This can come in two forms. The first is that the office owns the property. In the case of the Emperor this means the Imperium. The second is that the person owns the property. Emperor Strephon holds own dozens of worlds outright. Some as the Emperor of the Third Imperium and others as Strephon Allea Alkhalikoi. For example the Depots are owned by the Emperor of the Third Imperium and are assigned to the Imperial Navy to administer under the supervision of the Domains. The planet of Calis in the Core Sector is a resort world owned by Strephon Allea Alkhalikoi and is run by the Calis Resorts Limited with the dividends being paid directly to the Strephon's household. The same technique is applied by Dukes, Counts, and down the line.
This means while the Imperium is highly decentralized the Emperor and/or nobles often have considerable local resources in various regions.
The primary effect of my Imperium on the game I ran is that the Imperium comes across very disjointed and sluggish to the players. They can literally run rings around their opponents keeping one step ahead of the law so to speak. Until...
Until they draw the attention of the nobility. The higher the level of attention the faster the Imperium start responding to the players. Of course this means there also an out for the players. Get a noble on their side (or the player be a noble) then the Imperium returns to being sluggish.
All of this really only applies to hopping between worlds. What happens on a particular world is pretty much the province of that individual's world society.
I hope this helps with your version of the Third Imperium.