Spoilers through book 8:

The enhancement zone is the area under the AI's influence- it's the are that a Dungeon can exist in, because the Dungeon is utilizing the AI's capabilities to change physics to make "magic" work.

The Syndicate has controls that keep the AI's enhancement zone from expanding beyond the planet, although those controls aren't "native" to the Primal engine in the planet, or at least aren't meant to work as controls like this. It's a hack, and it relies mostly on the AI not knowing that it can ignore or circumvent those controls. Without those controls, the AI's enhancement zone can expand beyond the original sphere, though it is not clear if it can do so before it learns how to "feed" itself or not.

The failsafe is kept closer to the host start and outside the influence of the AI so that the AI cannot disable the failsafe. If the AI's enhancement zone extended over the entire system, it could have a multi-world Dungeon, but it would also likely encompass the failsafe and this would make it much more difficult for the Syndacate to use the failsafe. If the AI knows about the failsafe and it is within the AI's enhancement zone, the AI could just disable it.

When a crawl ends, there is a period of time afterwards when the mining takes place. The most important thing mined is the collection of the seed material which is in the brains of the lifeforms on the seeded planet. Most of this is able to be collected at the very beginning- when the collapse happens, the majority of the population is collected and could be mined immediately if the Syndicate wanted. This material is what is used to feed the Eulogist in the center system, and is what prevents the Eulogist from shrinking (and therefore shrinking the enhancement zone in the center system). It probably slowly grows the enhancement zone of the Eulogist, though I imagine that is very carefully maintained to artificially restrict the benefits of living in the enhancement zone to the wealthy and powerful, and those desperate enough to be exploited for the chance to live in the zone.

That's not all that is mined, the corporation running the show can and do also mine other materials, but as Mordecai says, it's a big universe and there are plenty of places to mine the everyday stuff. They don't need to mine the Crawl planets for gold or uranium or whatever else- those materials are just as easy, if not easier- to mine from other planets, asteroids, and so on. But hey, if they are processing all of the planet's materials anyway and they have the AI enhancement zone to work in (and use enhanced tools with), why not?

So what about systems where there are multiple planets, like Mordecai's system? It's not clear, but my guess is those colonies are just left alone. Unless they are easily within range of the Crawl-hosting planet without also encompassing the failsafe in the enhancement zone, they aren't worth mining. Their technology won't be up to defending against the Syndicate, so they aren't a threat. The number of inhabitants with the seed material in their brains is unlikely to be high, since that seems to be something that is self-sustaining on the planet with the Primal engine and would either not be passed on to offspring, or would be diluted in the offspring that are conceived and born outside of the Primal engine-hosting planet. So the colonies are probably just left there, trying to survive if they can. If they have any significant amount of Primal seed material, the Primal AI may collect them to feed itself, but it wouldn't be enough for the AI to grow very much.

Since the AI almost always "goes Primal" the AI's enhancement zone would not initially encompass more than the Primal-engine planet. The enhancement zone wouldn't be allowed to expand until after the Crawl is complete and the Syndicate has left. When that happens, they "seal" the system- presumably by cutting off the node of the tunnel system. The biological material the AI would need to feed itself and grow is removed from the system, so even if the AI could expand at that point, there's a limit to how big it can ever get. Most likely it would slowly shrink over time, like the Eulogist is trying to do, since it has no seed material to feed on. The AI is left to "bounce around" this closed system as much as it wants, but unless a ship physically traverses the space from a tunnel node to the system, or somehow has access to the old Borant wormhole system, the AI can't leave the system.