Military officers are arguably the most coveted applicants -- by adcoms and MBA recruiters alike.
The reason why the numbers may seem small is that there are relatively few applicants from the military that apply to b-school - compared to other folks (bankers, consultants, engineers, other corporate professionals).
Doesn't mean that every single military officer gets in, but they have had better success at getting in than other folks. From my memory having come across many applicants in the past, the few folks that I've seen get into all three HBS, Stanford and Wharton tended to be military guys.
As far as I know, the adcoms on the whole don't have a preference for one branch of the military over another. It's really a function of the applicant pool itself - if one year there's more Navy guys applying, there will be more Navy guys in the class that year. Having said that, I have come across quite a number of exceptional guys in the special forces units, and those doing multiple tours in a combat role.
In terms of the admissions process, the most common hurdle for military officers is academics and/or GMAT -- historically they tend to have weaker scores than average. However, if your academics and GMAT are in range (3.3 and higher, 680 and higher), then it really comes down to how well you execute your application and showcase which aspects of your leadership/team experiences in the military transcend the military (i.e. being able to show that you understand what aspects of your leadership experiences might be applicable to the civilian/corporate/b-school world, and what aspects are military-specific). In other words, what the adcoms are really looking for is that you as an individual understand what it takes and are fully prepared for the challenges of transitioning back to the civilian world. Having said that, adcoms like many civilians on the surface find those working on the front lines in a combat or intelligence role more compelling.
With recommendations and superiors: in my experience with military applicants, none of them had any issues with their superiors and most if not all were extremely supportive of their decision. HOWEVER, timing is everything - bring it up the year you intend to apply (usually close to the end of a 5 year tour), rather than revealing your intentions so early on.
As for post-MBA jobs, the veterans with MBAs network is tighter than the school networks themselves - and they transcend the schools. In other words, it's a pretty tight knit and helpful group across the top schools no matter where you go (those who go to Wharton know the guys at HBS, and who then know the guys at Darden, etc.). Whether on a trading floor at a bank, consulting firm, corporate office, etc. being a veteran can be a big asset especially if the hiring person is a veteran himself.
As for how civilians think, I don't think you'll come across anyone in the business world that does not admire, respect or value military experience, regardless of their political views. Again, the one thing that a veteran has to do is to convince the recruiter that they are ready and equipped to make that transition into civilian work as smooth as possible - that even if the recruiter himself/herself doesn't understand or appreciate the actual differences between the military and civilian world - that you are able to convince them that you do so that you can get slotted into any project group as seamlessly as any other banker/consultant/engineer who is accustomed to a corporate office environment.