A commenter pointed out that it might be awkward for the interviewers to ask about the asylum notation. I agree. There are many restrictions regarding what we can and cannot ask about in an interview - including age, marital status, national origin, etc. I would be concerned as an interviewer about asking the candidate regarding his or her asylum notation out of concern it might run afoul of employment laws and regulations. To me, this seems like something you could address in an interview. Like where an interviewer asked about difficult challenges you have faced, or unique circumstances, or why you went to law school. I think in that context, your asylum situation would be an interesting and unique take on those questions.
As for other things in the "personal" section, I don't usually spend too much time looking at it; honestly, I would really only remember if it had something unusual (or odd) - which wouldn't necessarily be a good thing. Most of them say similar things - cooking, travel, road races, etc. I am more impressed when people show me deep experience and an ability to juggle - since that is what we do every day in law firms - we typically don't get to work on one memo all day or for multiple days until we feel satisfied; rather we have to put out multiple fires and oftentimes when we think we will spend the afternoon doing one thing, a separate unexpected issue comes up and totally derails our plans.
So, if you've supported yourself by working through college/law school - that is something I think should be noted. If you served in the military - say in Iraq and commanded a unit -- yes, that is relevant and important experience -- clearly you could operate under stress! (And yes, I have seen reference letters from commanding officers that did help a candidate get a job).
If you've had unusual and challenging internships, note those as well. I would just stay away from too much personal stuff. Are you moving to a "new" city to be with your fiance? Not for the resume, but more for the interview or perhaps cover letter to explain your connection.
I prefer one page resumes, but if you've had a LOT of work experience -- i.e., you took time between college and law school of more than a couple years -- and you feel you need to go to the next page - then go ahead. But bear in mind some reviewers may not turn the page.