Yes - you need to have 1 clinical year of experience to apply to PM or OM. It seems like most do IM, peds, or FP, but you could do any prelim or a transition year. *Some* programs require that you finish a full residency first, but those are by far in the minority.
Yes - it's not that uncommon for people to have experience in something else, even for several years. I think there are even people who have done pathology. But, you still may not want to wait too long from the rest of your medical training. You still want to have fresh clinical skills, and you have to have completed all the USMLE steps to get licensed (probably not an issue if you were working in another medical speciality, but may be a consideration if you were doing something non-clinical for several years).
I'm not sure how to answer the question about how competitive that they are (compared to other disciplines). Since all (?nearly all) PM or OM programs interview people who are "PGY-1" or more senior, they don't participate in the match. So, the data comparing recent med grads in other fields isn't readily avaiable (as far as I know). That being said, I don't really know many people who wanted to get into the fields and who were not able to find a spot that they were pleased with. I even was checking out programs "off cycle" and some places offered me interviews (I think I was inquiring around April for spots that would have started in Jun/July). There aren't that many programs and spots... but there aren't that many applicants vying for those spots either. Unfortunately, the funding for these spots is also not tied to CMS (Medicare / Medicaid), so things seem precarious (at least, to me). But, most argue that we need more PM docs (particularly with the Affordable Care Act). OM has slightly different funding, and definitely has more spots / programs. And, in both fields, there are some programs that are clearly more competitive than others.
On finding work after finishing... ACPM (I linked to them above) has job boards and email bulletins that I get. There are a lot of OM job announcements - in a broad array of communities. PM spots are a little less obvious... but, being dual-boarded (say, IM/PM) can add a lot of job security.
I can only speak for my program with authority, but it seems to me that a lot of PM/OM residents are happy, enjoy their work, and don't really have that mentality of being in constant competition with their peers. But, I'm sure we all work hard, too. In my program, you have to be good juggling your time. In most programs, you earn an MPH, so you have to be comfortable with graduate public health classwork (my opinion is that it's not too hard, but this is clearly in my interest area).
