The first question, I think it's because she doesn't know where Shinpei went since Shinpei never revealed his name to anyone when travelling. The second question, is because in his heart, he believes that she's already dead. The third question, he couldn't bear to see his beloved got buried if she did die so he left
Well let's imagine you in her shoes, your beloved was found gone after your recovery. You tried to find him but your rich friend cannot detect him anywhere, that's why you wrote in your letter "Are you on a journey or perhaps living quietly in the mountains?" because you're unsure where he is since no one he met knows his name and only knows his deeds

I would rather it is a tragic ending...
the plot twist seems to have been a last-minute decision to turn it into a happy ending, without any logic.
there are so many loopholes: why didn't satoko find shinpei san as soon as she recovered? why didn't shinpei san visit even once during the 50 years? why did shinpei just left satoko's dead body in mitsuharu's place without even burying her?