I really like this idea. And, of course, so predictably, I have some to add. Yay!
For FP: Territory gives. That's your incentive to expand: that's the ONLY way to get FP. Having invulnerable stations drains a constant amount of FP. Say, 3 systems/station claimed would give you a net FP of zero. This number can be anything reasonable (or unreasonable); however, you may also need limits on station size. These don't have to be hard limits; for example, limiting asteroid spawning around massive stations, or some such. You need reason to place multiple stations in systems. Perhaps a docking limit, on non-turret docks, or a docking mass limit? You can only dock so much mass to a station with so much mass; say, 75% or maybe higher. Probably 85% or so; stations can no longer support infinite numbers of capital ships on a parking stick with attached open-air factory. Although, perhaps an exception factor is in order, for stations with protected docking and no inclination to add an A-armor block.
Taking hostile territory would be necessary because, given you don't want to spend hours flying in and out of raids, you can't build on the far side of the galaxy, yet you still need territory. Perhaps if you can't move your HB without removing the station. This would make it more likely for groups to expand around their HB, given that it cannot be moved.
Assuming you want to defend it and hold it as a powerbase, it needs to be in a position to control as much of your space as possible. In practice, this means a spherical (ish, we're dealing in cubes) expansion is necessary to absolutely conform to this rule.
The reasoning is simple: the HB provides a safe base for a response fleet, be it a fleet or player ships, because it cannot be neutralized, no matter what the opponents bring. Thus, you want your response fleet close to your borders where (keep going for the explanation) the enemies will attack.
And all this is to incite combat. For instance: if you have good reason to expand, and not remain idle; and to take hostile territory for additional mining territory (assumes no extravagant bonuses and a desire for big ships) in a defensible pattern around the immobile HB (See Note); and to defend that territory to the death and destruction of expensive combat vessels; and reason to defend stations (shops and trading - cheap cargo vessels made able to earn good profits due to different prices across the galaxy), then you will, of course, defend that area - for it is more valuable than your ships and death.
Also, death FP cost should go away. It'll help with the whole Urging-People-to-Interact side of things. Including hostile interactions. Plus it takes away the FP points that need to become rarer to mean much; on default settings it's too easy to build up tons and tons of points forever. I think player points should stay, but they should gain maybe 25 (as lost at death) for every hour ONLINE; not "Active" within days. AFK doesn't count. Logged-in circling bots should not count.
Also, asteroid resource returns may need to be reduced, given that we're trying to establish an expanding game structure, where turtling up is possible, but not really beneficial, and expansion is the way to gain power. And that's the point. Plus once NPCs are really introduced into the game, we'll have some reason to expand: perhaps crewer recruitment from limited supplies of willing slaves, or NPC resources (i.e. TRADING! with different prices in different ports) that are limited by the number of NPC locations you defend/control/have access to.
(Note: hey, maybe a specific Home-Base block that must be moved? there's an idea! It can go in cargo, but not be bought, and faction members can possess only 1, ever. Lose it, and you're done, because either you've been caught with your pants down and it's in transit, or your HB is gone; it can be generated when the HB reaches X mass, and then transferred to any other station above X mass in your faction, that you wish to be HB. Before it reaches that threshold of mass, it's invulnerable and protection is NOT transferable.)
Edit 1: Cleaned text wall.
Edit 1.33: Cleaned text wall.
Edit 1.5: Fixed ideas and consolidated information into blocks.
Edit 1.66: Cleaned edits.
Edit 1.999999: Just kidding!