Will the debris fields and such disappear after a bit? Otherwise I'm worried that the pirates will win anyways through OoM errors and super lag
They will despawn, yeah. In the "map after pirate raid" screenshot, the derelict ships haven't despawned because the player fleet is still in-system.
Do bases serve economic purpose as well? I assume so, but you can land/trade etc there just like anywhere else?
You can actually land there and trade, if you manage to avoid the fleets, but I'm not sure how I feel about that. It's mainly a consequence of those being set up as proper markets on the back end.
Could you form an alliance or up relations with pirates and then use these bases, or smuggle to them, or have your station raid them back?
Just curious what options there will be (and for some of these I'm more curious if that's an eventual maybe, not next patch stuff).
At this point, no. There are some possibly interesting options there but I don't want to delve into it without getting a lot of other stuff done first. Breadth first, then depth as needed, at least for this stuff.
Can new pirate bases spawn (either spontaneously built or stolen from someone), and can old pirate bases destroyed rebuild elsewhere?
Yes they can. "Old" bases don't reappear; a rebuilt base is a new base as far as the game is concerned.
What about playing as pirates? Can we build pirate bases and attack colonies too? I love being a pirate lol
I'm thinking through the options for players raiding core colonies; no specific comment quite yet

You wouldn't be able to build exactly the same sort of base as the pirates - which is just a standalone station - but you could certainly build a colony with an orbital station.
Are pirate markets like Barad and Kanta's Den still in the core systems, or have they been pushed out into the random fringe worlds where the bases are?
They're in the core at the moment; not 100% sure on how it'll end up. Barad had to lose its orbital station defenses, since that kept attracting the Hegemony patrols and getting destroyed, but that... kind of makes sense. A Hegemony administrator might turn a blind eye to some "perfectly legitimate miners", but that's a lot harder to do if their station is bristling with guns.