On the first question, it depends on what you mean. Science can answer physical questions about physical, measurable things. Science can tell us that, when we die, our bodies begin an immediate process of decay back to the basic elements. Science says that, death is truly an end of life. If science offered the only possible answer to the questions of the universe (it does not), then science tells us that when we die, we are dead like Rover, dead all over.
However, almost by definition, science cannot answer questions about spiritual things and, presumably a soul or a spirit is not a physical thing. Science is simply not helpful on this question (even if some unbelieving scientists try to turn science into philosophy and will try to tell you that they have the final answer)
Do we have any evidence for the existence of a soul or a spirit? That depends on what kind of evidence you are looking for. If you are asking for scientific evidence, the answer is a solid no. An early nineteenth century scientist John Dalton asked that he be weighed immediately before and immediately after he died so that it could be determined if he lost a small amount of weight. This weight, he proposed, would be that of a soul. The experiment gave the null result. Souls/spirits, if they exist, appear to have no mass. This should not surprise us, but you cannot blame Dalton for wanting to do the experiment.
But we have biblical evidence for the existence of souls. There is plenty of such evidence (although some materialists will scoff at such “evidence”). Jesus, who walked on water, created bread and fish, calmed storms, healed the lame, the blind and the deaf and raised people from the dead; the same Jesus who fulfilled dozens of messianic prophecies. This Jesus believed in the existence of souls and spirits. He raised Lazarus on the fourth day after his death, seemingly proving that there is in fact some sort of life existing after death. In 1 Samuel 28 Samuel spoke to Saul from Hades years after he had died. I could give a mountain of biblical evidence for the existence of souls. Put it this way. If it can be shown that Jesus is who he said he is and if it can be shown that the Bible is inspired by God, then souls and spirits are proved to be real. However, remember that this is not scientific evidence, and that some people will reject such evidence a priori.
In summary, the scientific answer to your questions is not the only and final answer. Science cannot answer questions about morality, value, purpose, ultimate cause and much more. In fact, arguable, science cannot answer any of the most fundamental human questions. Science, taken as the only source of truth says that death is final, but other kinds of evidence points in a very different direction.