Well, the dorms seemed pretty complete (at least the ones listed in Nagoya University), despite their very small size. I am still trying to figure out how one can have a bedroom, a kitchen and a bathroom in 14m². So, even in a dorm, I'd still be living by myself, and social interactions would be limited to common areas only, I guess. Could anyone with experience in this area confirm my thoughts?
It depends on the dorm itself. I actually lived in one that was like its own apartment, and it didn't have any set areas to meet with people. Subsidised rent was 60k, but for regular students it was closer to 100k (located in Kawasaki). I had a scholarship of about 80k a month + a two-day part time job and I lived pretty comfortably, but I don't eat or go out much! There were other dorms where you just your bed/desk and shared for kitchen, showers, laundry, etc, and some where you had roommates and didn't get your own space. You also have to think about things like curfews and whether you're allowed guests over - it was broad policy that everyone had to be back by 12am and no guests, ever, but my dorm manager was really lax and let us do whatever we want. Some of the other dorms weren't as nice and people did get locked out if they stayed out past curfew.
For Keio at least, when I applied as an undergrad exchange student, they let us know when dorm applications were open and we had to send in a list with our preferences - first in, best dressed. For the September/October semester, it opened around July, so I'm sure there will be plenty of time even if you find out the second screening results in February for an April departure. I know that some other universities don't let MEXT students chose their dorm (as in, they have one dorm for MEXT recipients), and others that might not let you stay the whole time because they need the rooms for other students (and in the case of subsidised rooms and private universities, they'll probably kick you out so they can get full paying people in haha).
And for the make-up of Japanese rooms... don't think of them as 'bedroom', 'kitchen', etc. Most of the time in this kind of set-up the kitchen will be in the hallway which leads to your single large room - the 'bedroom', but also your main/only living space. Bathrooms are quite small and basically are big enough to fit in what they need to. It's compact living, but for someone who isn't really in to fuss and likes to have just what they need, it's a pretty good set-up!