Windows shows a warning when I open the installer
That's the usual SmartScreen notice for apps downloaded outside the Microsoft Store. Choose “More info” and then “Run anyway”. Always download from this site so you know it's the official installer.
Do I have to update by hand?
No. The app detects new versions and updates itself. If you want to force it, download the installer from here again.
Windows asked for network permission the first time
That's expected: Magic runs a small local service on your own PC to deliver the overlays to OBS without going through the internet. That service only listens on your machine.
Can I install it on more than one PC?
Yes, your account works on all of your computers. You can even aim Minecraft actions at one specific instance or at all of them at once.
What if I stream from my phone?
You use Magic Mobile, the native Android app. Same account, with its own profiles and its own separate subscription.