Question 1 What is the role of applications in modern mathematics ?
I don't know if there is such a thing as THE role of applications, but I think it's a great motivation for developing a theory and/or solving a particular problem if it has an application outside of mathematics.
Then again, the same can be said about applications to different branches of mathematics. Besides, applications are not the only source of motivation to do mathematics. Perhaps, one thing real-life applications play an important role in is to make it easier for non-mathematicians to see that mathematics isn't pretentious junk art with no practical or intelectual value. This isn't to say that mathematics with no real-life applications is junk art. But if it has a practical application, it'd be a lot easier to defend your own work from such criticisms from outsiders.
Question 2 Different countries have different mechanisms to stimulate interaction between mathematics and applications - what are these mechanisms and what are their advantages and disadvantages ?
I don't know much about other countries and other branches of mathematics, but it appears that in Japanese universities discrete mathematicians tend to work at non-math departments (typically but not exclusively departments of computer science or computer engineering) slightly more often than in some other countries, regardless of whether their work has immediate applications to the respective fields. And when I say discrete mathematics, I include some branches that some may not consider discrete mathematics, such as number theory and algebra.
From graduate students' perspective, this is a great thing because you can work on "mathematics for the sake of mathematics" if that's your thing while naturally getting a lot of exposure to the applied side of mathematics. For example, I finished my undergraduate study at a traditional mathematics department and went to the graduate school of mathematics of the same university. Then a couple years later, I transfered to the information science department at a different university. I didn't change my research topic during my Ph.D. program, but this transfer definitely influenced my view and attitude toward mathematics and gave me a lot of opportunities to learn stuff outside mathematics, which I don't think would have occurred, at least not to the same degree, if I stayed at the math department.
The negative side effect I hear from faculties is that it makes it harder to get graduate students with strong backgrounds in mathematics because math majors tend to apply for math graduate programs like I did at first.
Question 3 (for pure mathematicians)
Hmm... I don't know how you define that pure mathematicians thing, but looking at my own publication list, I guess I'm not pure or innocent anymore. Oh, well.
