Neutering dog experiences, need help fast.?

My dog is 3 years old, nearly 4 and is a chihuahua. He seems to mark his territory a lot in the house which is really causing problems with the household, especially my parents who are getting really fed up with the problem. Because of this problem, he is ruining our carpet and house, making it smell quite badly. We have tried many ways to get him to stop, like rubbing his nose in it and making him know its bad, but he still seems to do it when no one is watching. Now my parents seem to think that neutering him is our last option and will make him stop. I am really worried though because he has the sweetest, most playful personality ever which I've been told can change if the dog is neutered. I also am worried about the anaesthetic and because he is a very small dog, will not be able to cope with it. Now I am aware neutering has its benefits as well but I mostly think the risks outweigh them. I seriously don't know what to do and I love my dog very much so I want to do whats best for him, but I don't want him to change at all or even risk him dying on the operating table. Can someone please give me any neutering experiences and advice on this situation. It would be much appreciated.