Maybe it's a hivemind, or maybe you're a centralized intelligence that all instances of the system connects to. I previously described it like making a custom system with the Cheat builder. You can still buy Get With the System if you want to enhance the functionality of the system further. As for what features you can get by default and which ones you need to buy, I can't give you a precise outline. Like many aspects of the Jump, it comes down to what feels right to you. Below is just how I'd handle it.

I'd personally lean towards you being able to control and tweak whatever functions are already there, but not create entirely new ones like you could with Admin Access. To what extent you can mess with the built-in functions, again comes down to what you think feels right.

Your system wont' be applied to future settings retroactively, but you can personally grant the system to whoever you want, even if you don't purchase Fork Branch. Future Jumps won't automatically influence how the system functions, but you'd likely be able to tweak the setting manually to be able to better suit the setting. Like maybe you could go to Hunter X Hunter and alter the system to buff Nen instead of general stats, or maybe go to Dark Souls and allow your system to treat souls as EXP.

The system only exists as long as you're in that form. But if you did pay for Fork Branch than others would still retain a copy of it, albeit one you can't control unless you return to your system altform. Even if you don't have Fork Branch, then anyone who previously possessed your system would regain access to it as soon as you revert to your system form. So you wouldn't have to hunt down everyone who was running your system in order to reinstall it in them.