Why is my mood dictated so much by how well my research is going?
Does anyone else experience this?
Yes, I'm pretty sure we all do to some extent, and that it's one of the defining characteristics of being a researcher that you get so emotionally involved in your work and are so passionate about it that it has that effect. It can be both a curse and a blessing (see the fantastic question linked to by jakebeal in the comments). Hopefully if you are talented and work hard, it will be the latter more often than the former.
As for how to manage it:
get used to it (I mean that literally, not as a sarcastic admonition) - if you plan a career in academia there will always be ups and downs of this sort. At some point one learns to be patient and not to freak out every time the research isn't going great.
Always have other more "normal" work to do, like teaching, that can help you feel like you're doing something worthwhile even during those times when you're stumped with your research.
I agree with others that this is natural, though I'm not sure I would say its unhealthy. It's how you deal with your emotions that may be healthy or unhealthy. Here are some suggestions, which have some overlap with other answers.
First, realize that your goal is to learn and understand things, not to get a result today. The world's not going to come crashing down if it takes you months longer to solve something than you hoped, or even if you never solve it. This can help you to enjoy the full range of research, of which "struggling in the dark" is a large part. Barry Mazur's fond of saying "when something is hard, that means it's interesting."
Cathart. Talking to other people about your frustrations can help you deal with them, and also give perspect. (Of course, you don't want to vent all the time.)
Try to find other things to do so that you feel productive. This is one reason it's helpful to have multiple projects going on at one time, preferably each in various stages, which is common for senior researchers though less common for junior researchers. Maybe you can spend time writing up notes on something, reading papers, or learning a relevant topic, to at least feel like you're accomplishing something.
I personally have not mastered separating work life and home life either (I'm also not sure it's necessary), and sometimes I come home down about work. I find it helpful to have an engaging non-academic hobby, to get your mind completely off work for a time. It could be something physical like sports, or less physical like novels/cinema.
For me, the simple attempt is to achieve at least one positive thing a day and hang on to that. If all else failed, I still have something that I can feel happy about. It might be as small as automating some mundane task that I repeatedly did manually, preparing a couple figures that will be useful for some presentation/publications, etc.
Obviously this assumes you have multiple parallel things running, but that is always good anyhow. (i.e. "not all eggs in one basket")