macOS Internet Recovery is a NetBoot-like service. Basically, a bootable image is downloaded via HTTP (from an Apple or an Akamai server) into RAM and your Mac boots to it afterwards.
So yes, you can run First Aid on the OS X Base System volume and it will not erase, restore or alter partition layout because it's not a writable volume.
Since it has to be ...
If your Mac originally shipped with Mavericks, then yes, you can install Mavericks on it, regardless of whether when you bought it, it had Sierra on it.
Otherwise, no. Think about it. If your Mac was released after Mavericks, Mavericks won't have hardware drivers for it.