Have you tried installing it with Windows 7 compatibility mode?
You can also try a workaround if you have Windows 10 Professional. Activate the Windows Sandbox in the 'Turn Windows features on or off' dialog. After that a Windows Sandbox app will be available. If you start that, it will create a virtual machine and connect to it via remote desktop. This virtual machine doesn't have any protection enabled. It's for testing apps. So, try to installing A7 there by copying and pasting the installer and the key file. Enable Windows 7 compatibility mode for the installer. After that, simply copy and paste the installed A7 folder back to your computer.
I just tried it and it worked. I got an error during the part of the installation where the installer tried to install the Visual C++ 2008 Redistributables, but A7 was installed fine. I copied and pasted the GStudio7 folder back to Windows and ran an example. Works without issues.
I don't think that your key file disappeared from the CD. It more likely be that Windows Defender or your Antivirus just quarantined certain files. You can reverse that in the antivirus settings. This may be caused by Gamestudio's code compiler. Maybe your antivirus doesn't like that.
If you want some kind of modern version of Gamestudio you might want to check out Leadwerks 5. It's currently in early access. But maybe you know that already.