It's like if you're making a state within a state.
Houses have their own way of functioning. The BPA take care of the laws, like special cargo permits or POB permits, and they also intercept smugglers and escort merchants. Using their TLAGSnet they can see which ship docks which lane/gate in Bretonia, and scan its cargo using the /showscan [name] command. However, they can't fly battleships and don't have much interest in raids. They also can't fly large transports.
The BAF defend the House from its foreign enemies. That's why they can fly all the military ships. However, they don't have any business in approving, for example, base construction, and are only the strong arm if the base is illegal. They don't have TLAGSnet access and can't fly large transports, but can still uphold the laws and fight pirates, although that's mainly the BPA's job. The BAF handle foreign capital ship permits.
The BMM are the national industrial company. They produce ships and equipment for the two above, themselves and Bowex, as well as any civilians who can pay for them. In return, the two above protect them and Bowex ships their stuff. Their main business is mining. They have nothing to do with conflict, as they are civilians, but they do have some sort of security (like banks iRL have security, which has firstly been government-approved), hence they can fly fighters, bombers and gunboats. In real life, a company that produces tanks sells them to the military, and doesn't use them itself.
Bowex is a national shipping company. The BMM mine out the ore, smelter it, make stuff of it, and Bowex export the excess and thus make money for Bretonia by trade. In return, they ship back the resources which Bretonia is low on. The BMM process them and use them for producing things.
When in Bretonia, Bowex is protected by the BPA and the BAF, and supplied with ore by the BMM. When in a different House, for example Rheinland, they are protected and supplied by Rheinland's organizations. They have nothing to do in, for example, Omega-41, but in case they just need to go somewhere very far, they can use gunboat escorts of their own.
Gateway are a private shipping company. They are not so tied to the government, which means that they are slightly more free in choosing who to sell goods to and where to go, but in return have less say in the Kingdom's politics. Of course, all embargoes to foreign factions apply to them as to anyone else. They are just civilians from Bretonia, making money for themselves and abiding by the laws, being taxed as common citizens all the while. They have some sort of security, of course.
All these above form the Kingdom of Bretonia as one faction. Every one of these has one or two representatives in the War Cabinet, which currently functions as the government of Bretonia. As you can see, each is specialized for one thing, which makes them the best at that. The other thing is covered by the others, who are best at their jobs. Because Freelancer is a shooter game, we should separate the combatants, who are the BPA and the BAF, and the civilians, who are everyone else. I have yet to see a civilian operating a combat-ready battleship or a military to trade oil with huge tankers.
I hope this makes it clearer how Houses work.
PS: I forgot Planetform, but they aren't official anyways. But you should get it by now. I wish you luck with your officialdom, and I suppose the BMF can get a representative in the Cabinet if you succeed. I've been wondering though, what makes the faction different from BMM? Maybe it's more of a private company, like Gateway?