I guess that your immune system would be more active than it should and this would be picked up as it is part of most tests? and investigated further if something unusual was found.
Yes. Generally they won't find it because they're not looking for it, but if someone has cancer in their blood it wouldn't matter what purpose the doctor took the blood for.
I'm pretty sure they have to be looking for cancer specifically to be able to detect it. I don't know if cancer cells are easy to spot in the blood.
It's all a question about what the lab is testing for.