What is the worst that can happen to a player colony if the player doesn't do anything? Would it only affect stability or can our stations actually be destroyed? I'm a bit worried over how much babysitting we'd have to do (pirates + whatever else might threaten your base), but I suppose once you upgrade your station well enough it won't be an issue.
If a station is defeated in battle, it goes into a "disrupted" state (a general thing for industries and structures) for half its build time, after which it comes back. This doesn't cost anything, but fo course the station is unavailable for combat support during that time.
As far as the worst that could happen - well, it depends. I'd imagine pirates would be interested in raiding, not destruction, though if they were particularly angry at you for whatever reason...
Let's just say that - if we expand this beyond pirates, also - that all options are on the table. The more severe outcomes would need to be more rare and signalled clearly, of course.
Also: I feel like in many of the responses, "pirate raids" are assumed to be a common thing and the main thing that pirate bases do. That's not the case; the main impact of bases is the "pirate activity" market condition, which is just a passive debuff which you can deal with by destroying the station, or by just ignoring and accepting the hit to your colony stats. Raids, which also don't exactly require player intervention, would be more rare.
Generally, the idea is that there are a lot of things for the player to deal with, and they have to prioritize what to deal with first, and what to ignore entirely until some future, more convenient time.
Is there a strategic-level AI governing the response to raids and counter-raids? Like, if I drop into a system unnanounced with an Onslaught for a doorknocker and a pirate fleet spots me, will they gather nearby fleets and attack me? Pull back all fleets in-system to defend the station? Or, if the forces are lopsided enough, simply abandon the system? (I better still get paid in the last case)
Will this change if I have my transponder off? (Maybe I'm just coming to trade!)
Bases don't shut down, no - I think that might just create problems in the name of doing some "realistic". There's some level of strategic AI but it's not particularly involved. In particular, it wouldn't do any of the things you say, but it would defend the objectives. It could be pretty easily extended to defend the base as well; basically it just creates a set of points that need to be defended, each with a weight/expiration time, and patrols/military fleets distribute themselves accordingly.
If the player leaves a pirate base alone long enough, will it ever go away on its own? Might one of the major factions could mount their own expedition to destroy the base?
Might the player gain the benefits of a faction's objectives through a commission or some other channel?
Maybe at some point? These seem very much like "optional extras".
Overall, these new game systems look like they'll add a lot of depth and interest to the game! Definitely excited 

1. If players don't have strategic-level control of fleets and can't take command of fleets remotely, how is the game going to avoid becoming Whack-A-Mole?
1) system defenses, 2) raids actually not being all that common, and 3) mostly not that disastrous. It's just a thing you have to prioritize responding to vs something else you might be doing.
2. If build-times are a thing, then players will get stuck defending every single Outpost until it has Pirate-proof defenses, most likely. Won't that be pretty boring? Shouldn't players be off doing something fun and letting their underlings get on with it?
3. What happens when players are, to put it gently, at the point where these Pirate groups aren't scary to their fleet at all, but can still (somehow) do relevant damage to their interests?
Again, see point re: raids not being super common. I'd also imagine pirates wouldn't find out about a brand-new colony or raid it immediately. As for the later game, I'd imagine system defenses would be strong enough at that point for minor raids not to matter.
5. Shouldn't a player be able to bribe Pirates into raiding elsewhere? Paying barbarians to harry somebody else's borders is almost always better than fighting them, IRL.
Did think about it; another "optional extra" that might be nice at some point. On the flip side, it's got the "optimal play avoids combat" issue if it's cheap enough. Could possibly be an over-expensive option that you might take if you're in a bad position otherwise and can't respond to everything.
6. Shouldn't player patrols discover Pirate bases being built in nearby Systems?
Was thinking that you might discover the location of a base for free at some point, yeah, if you've got a system with patrols nearby.
7. If that's so, why aren't they wiping out the bases during construction?
8. If this works for the Pirates, economically, why doesn't it work for players or Factions?
This seems to be coming at it from a "realism" perspective, which imo is not very useful. As usual, there are all sorts of reasons for why something might or might not "make sense". E.G. I'd imagine that whoever is constructing the base has enough firepower that patrols - which don't go out of the system much anyway - spread thin enough to comb multiple systems - would simply have no chance. Or maybe they do wipe some out.
I'm sure I could come up with plenty of other reasons as well. And I'm sure one could come up with plenty of reasons for why this - or anything else - *doesn't* make sense. Ultimately, it's up to the player to decide which they want to do. IMO, any sort of sandbox experience requires this sort of "cooperation" from the player, in terms of the story they're telling themselves.
The real reason is "it would be bad for gameplay".
How does the whole "pirate base spawns near player colonies" thing work out if the player colonies are in the core systems? First thought would be Penelope's Star, but could also apply to some randomly generated worlds in inhabited systems that might be colony-worthy.
Hopefully near the player colonies is still far enough away to not be silly in such a situation.
I guess it'd have to be a bit further away. Haven't gotten to coding this part yet, but there's nothing fundamental preventing pirate bases from operating from across the Sector if need be.
How easily and readily is the new system moddable? Let's assume that I want to make the Luddic Path use the same system for pirate bases. Could having multiple factions with pirate bases cause trouble such as excessive raids? If player is friendly with a Pirate faction, the AI probably understands that it does not need to spawn pirate bases near player's colonies, right?
(I am asking mainly due to my plans of making a submod for Nexerelin with various minor factions added by splitting the Pirate and the Independent factions.)
(Hi, and welcome to the forum!)
This is very moddable, yeah, though excessive raids might indeed be a problem. I'll see if I can make that tunable somehow, but no promises.
As far as the player being friendly with pirates, I don't know. In theory the code should handle it, but game-design-wise, that possibility is becoming more and more troublesome. Frankly, I could see just hard-locking the player-pirate reputation to -50 at some point. You don't need to be friendly with pirates to be a pirate yourself - or to deal with pirates - and since pirates cover *such* a huge range of possibilities, it doesn't make a lot of sense to be friendly with them in the first place.
(Also: I may or may not have plans to do something mechanically related with the Luddic Path.)
Can't factions set up small automated defense platforms to fend off players from the objectives? Otherwise I'm can imagine players spam transverse jump right after they've done sabotaging.
Hmm, this doesn't really seem like a problem - more like things wworking as intended - and a clean getaway doesn't require a transverse jump anyway.
That being said, I hope you reconsider the pirates and REDACTED not spawning together. That would be an amazing opportunity for the player to strategize and exploit the enemy. Perhaps leading a fleet of REDACTED into the pirate fleets to whittle down their numbers and make the fight with their station easier? I feel there's a more potential for fun and variety than otherwise. Taking on pirate stations will be super fun, but I would love even more numerous ways to do so.
I hear what you're saying, but what would most likely happen is just a bunch of non-stop random fights, with the pirates getting the short end of the stick. It wouldn't be the (neat) strategic thing you're envisioning.
Speaking of deserters, they do not feel like pirates at all. They feel like a major fraction with cheap red paint and a kiddie pirate sticker stamped on them. It is like a Hegemony fleet saying "We are not the Hegemony! We are pirates! No, really we are! <Right!>"
Again, "pirates" covers a wide range of possibilities. It's really anyone outside the law in the Sector, as defined by the major factions.