These questions are too broad, and not very clearly stated. First of all what's "an application"? Application to what? Is application to some "physics" which itself has little or no connection to real world (like string theory) counted as an application? We all know examples when this kind of "physics" stimulate a lot of interesting mathematics. Is an application in a completely different area of mathematics counted as an application?
Or only an application which brings profit is counted?
Anyway, no matter how one defines an application, my answers are these.
- Outside applications play an important role for mathematics as a whole. Probably as important as it was in the past. There is a huge (and not very well defined) part of mathematics which is called "applied mathematics". This usually does not include "mathematical physics" (at least in the US. But this is a matter of labels).
What goes under the label of pure mathematics, also frequently has applications, sometimes very important.
Second question. The mechanism that I know is financial. Those who distribute money want "aplications". In many cases they do not really understand what it means but they want it to be called this way. Various financing is available for applications, real and imaginary. Sometimes, there is an administrative pressure. But some money is usually involved behind it.
Third question. I qualify myself as a "pure mathematician". I define this as follows: the main criterion for choice of a problem to work on is usually aesthetic; I just like the problem. An alternative consideration is that a problem has potential applications in the real world. This is rarely a motivation for me.
Sometimes my results find applications in various areas, like material science, computer science (always unexpected to me). By "applications" here I mean that scientists from other sciences (who do not call themselves mathematicians) sometimes cite and use my results. I understand that parts of other sciences can be also very remote from the real world. But I am always pleased when people from outside use my results.
Sometimes an applied mathematician or non-mathematician asks a math question. I always try to help if I can. Sometimes I can. Sometimes the problem is even mathematically interesting. This is also very pleasant. In my youth, I was sometimes involved in "applied research" for money and other benefits. I did not like it. I'd rather teach to make my living.
