With your first question, I do think the university matters, and I'm sure the people in your embassy (mostly from the UK in your case) would know which unis have good programs in Genetics.
But of course, no matter what your grades are or whatever your university is, if your research proposal is not clear and solid, it would be hard to get past the interview stage.
Anyway, this is just my opinion, since I'm in the running, but we haven't received the final answers yet.
The order of your application is of course correct but that's not the only way. Being in the academe myself, I understand why some profs don't reply immediately as we're all busy. It wouldn't hurt to contact professors 2 years in advance, at least if anyone replied, you're sure to know they're somehow nice.
But you're likely to get more replies after passing the primary screening, since with a nomination from the embassy, they will really take you seriously. You have 2 years, the best thing to do would be to find out who the best is out there in the program you wish to enter.
And yes, English and Japanese proficiency only and nothing else during the exams.
Basically, it's all about the research proposal.
It took me a year writing mine.