Where do we start?

Yeah. I was using "subsector" more as size measurement. And personally I really like those traveller maps and divide for sectors and then subsectors. Doing them will really feel like discovering new universe...:D

I agree, I don't often use premade settings, but looking at travellermap.com really made me think about doing so for Traveller. But that is mainly because they a just so darn pretty.

I do worry that not having extensive maps could make speculative Trade hard to implement, but we can build out as much as we need when we need it.

Would these kingdoms also be controlled by a single faction? And have things like border patrols and border checks?

Well, this one has at least two factions, hence the civil war. Others will vary as well. We can work out the individual details together.
That would depend on many factors. Law Level would be a primary one, as would Tech Level, but there will be others, and if you use unconventional routes there will be ways to avoid such patrols and border checks (or to trade them for pirates, but that is the cost of doing such business:).