That would be nice. Sometimes there is good discussion happening that could easily die out because of people forgetting to check for comment replies on the other answer. There should also be an option to opt out of notifications when you're done with commenting on the answer.
Yes. I think it would make it easier to carry on conversations that do not occur on our own posts. If I comment on someone else's post, it would be nice to know if they commented back without having to go back and check every so often.
I think, although it sounds like a good idea, it would turn into a mess. If you post a question, then every opinion and comment should theoretically be about your question. If you post an opinion, every comment under it should theoretically be about your opinion. BUT, although every comment should be about that opinion, not every comment relates directly to the comments before it. For instance, you might comment on an opinion, then the next comment is still related to that opinion, but in a way completely different than your comment. Then a whole tirade sometimes follows about the coment after yours. The system has no way to know that they have nothing to do with your comment, and you would get possibly dozens of notifications about your one comment. Multiply that by many comments you make and you would have a boat load of notifications in which you are not interested.