The Sun's rays hit our eyes around 8 minutes after they are emitted from the Sun. Does this mean that the Sun that we see is always the Sun as it was some 8 minutes before? I strongly think this must be happening; is it really a fact? Do we always see the Sun's past?
Yes, you are right. We don't only see the Sun 8 minutes in the past, we actually see the past of everything in space. We even see our closest companion, the Moon, 1 second in the past.
The further an object is from us the longer its light takes to reach us since the speed of light is finite and distance in space are really big.
The speed of light in a vacuum is 299,792,458 meters per second, not infinite. Let's say, for example, particle of a beam of light, the photon, is emitted. It takes ~8 minutes to get to us; when it hits our eyes, we see it. This means that we see a photon that was emitted from the sun 8 minutes ago. We aren't, per se, looking "back in time", but we're looking at a photon that is ~8 minutes old.
In fact past, present and future is still beyond of our consciousness. In reality they exist all together as spacetime. For example, when you say, we are seeing 10 years past of a star which is 10 light years away from our planet, similarly an alien right now is seeing our 10 years past earth. A nice video here https://youtu.be/vrqmMoI0wks can be helpful to understand how mind boggling reality we are living in.