To respond to your first question: Yes, I think a significant other (or similar "family" reasons) is a good enough reason to want to transfer schools. For example, students might want to transfer because they need to be closer to a family member that is ill, or it turns out that they really cannot stand living in whatever city/place they're in now. I think being happy is very important to being productive and successful. However, you would definitely want a good (or at least positive) LOR from your current department/advisor for your application to school Y. So you will have to judge what is the best way to "break the news" to your advisor and department. If you feel that your advisor/department will "frown upon" your relationship, then perhaps you could consider not fully disclosing why you want to leave. You can cite "personal reasons" or whatever -- but it's likely that the more you are able to share with your advisor/department (within reasonable limits), the better/more support you'd get from them for the transfer (assuming that being able to share the information implies a strong rapport between you and the department/advisor as well as the advisor's/department's wanting to see you happy and successful).
For the second question: you are basically applying to the same school again, with what sounds like an almost identical application. Applicants change from year to year so maybe you could be accepted this time around. The fact that you are leaving a current grad program is probably going to reflect poorly without justification. In your shoes, I'd make it clear that School X is not turning out to be a good fit and you would really like a second chance at School Y. Personally, I'd contact someone in the department at School Y before submitting my application. But I'm not sure that is necessarily helpful -- it depends on what the people at Y are like! I think a situation like this warrants a "live" (phone or skype?) conversation to explain why you are leaving School X. I'd be sure to not mention "transferring", because you are not doing that -- you are basically quitting School X and applying to start again at School Y.
I also have a question for you! You say your relationship is not at the level that would involve career decisions. However, what you are considering here is definitely a major career decision. Sure, you might be unhappy at School X for lots of non-relationship reasons, and School X may well not be a good fit for you. But how do you know School Y will be?
I think it might be worthwhile for you to consider what you would do without the relationship. Would you still leave School X? If so, where else would you apply to, other than School Y? More importantly, if you are unable to consider/make a decision without factoring in the relationship, then it might be important to discuss it with your partner and re-evaluate the statement that your relationship isn't at the career-altering level, to make sure you are on the same page! Sorry -- I don't mean this to be harsh, just direct and hopefully helpful.
It's still relatively near the beginning of the year, and I know that many students, including me, experience self-doubt about the school choice and so on. So maybe after some more time, you'll find your fit in School X? Application season is approaching, but there's still a good ~2 months left.
Finally, many people (students and faculty) seem to give or have the impression that it's a bad thing to make career decisions based on "personal reasons" (e.g. family, location, significant other). I don't think this is a healthy attitude towards life, in my opinion. So I would encourage you to not feel guilty/shameful/bad/whatever on your decision to leave School X because of a relationship that you feel strongly about. I put equal weight to personal reasons and academic reasons when applying and choosing graduate programs and will continue to do so for all future career decisions. But you should be aware and be prepared for those that will "frown upon" students who choose personal reasons.