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.
Answer is Yes. Not only to sun and moon, we are seeing each and every distant object we are ever experiencing with certain delay. For example, if you are reading this text on a computer screen at 50 centimetres distant from your eyes, you are actually experiencing the past version of
One interesting thing to know is, in reality past, present and future exist all together as spacetime in the fabric of cosmos. For example, when you say, we are seeing 10 years past version of a star which is 10 light years away from our planet, similarly an alien right now is watching 10 years past version of our earth seating on that star.
An excellent explanation can be found here presented by Brian Greene to understand how mind boggling reality we are living in.