Angle of view is a combination of focal length and sensor size. If we're talking about two different sized sensors, to get the same angle of view, we have to adjust the focal length to suit.
Exactly. And you have to decide if we're talking about focal length or angle of view. You can't freely interchange the two concepts, because they are two different things. You can compare one to the other, but within limits.
Focal length is a property of the lens, and the lens alone. Whatever is a function of focal length, is not affected by the sensor in your camera, which - as you said - instead will affect the angle of view.
A Nikon 1 10mm f/2.8 lens will have an equivalent
focal length (read: angle of view) of a 27mm lens on FF, but it will be nevertheless a 10mm lens. Assuming transmission is the same, the exposure would be the same for this lens and a 10mm F/2.8 lens on FF. The Nikon 1 lens can be smaller because the sensor is smaller. The total amount of light gathered by a FF will be more, but the amount of light hitting the sensor / surface of the sensor
would be the same. You could mount the FF lens on the Nikon 1 and it would be the same - because the extra light will fall off
the sensor. That's why exposure is not affected.
If you now transpose this concept to this lens, you will understand that exposure-wise a f/1.8 lens is a f/1.8 lens disregarding of the system. This, again, is because f-stops depend on focal length and focal length is a parameter of the lens and not of the camera.