Does this gym have a back hyperextension machine?

Yes it does. It has both the one you can do with your own bodyweight as well as the machine where you sit and push back the pad with your upper body

It does have a machine for back hyperextenensions. It's not weighted. It's the type that you lean forward on at about a 45 degree angle and has an adjustable support at waist level .

I don't think so

No