If you've already destroyed the AI Command Station, then it is too late to perform any hacking actions in that system, sorry.
If you build you own command station there, you will take control of the Advanced Factory. You can then destroy your own Command Station to return the system to the Dyson Sphere, while retaining control of the Advanced Factory. Of course, this will turn the Dyson Sphere hostile for at least one spawn cycle, so you'll need something present to defend your Factory... In addition, you'll probably want to take and keep an adjacent system to give supply to the Dyson Sphere system, so you can build and maintain defenses there (turrets and forcefields, for example).
If you don't want to do that, then YES there will be another Advanced Factory out there somewhere. There is always at least one Advanced Factory and one Advanced Starship Constructor for each AI (ie, 2 of each) somewhere in the galaxy. So if you can find it, you can hack or capture that one instead.
You cannot kill the Dyson Sphere without also destroying the system and everything in it - and even then, only with one specific optional method enabled. This would seem self-defeating for your purposes.
As for the spawning, I seem to remember that the Dyson Sphere is more likely to be co-located with some high-value building, but I don't think the Advanced Factory is more likely than any other high-value capturable.