- The answer is basically tautological: it indicates that people who ask questions on Math.SE are less likely to ask PDE questions.
- Not really. There is a lot of general theory of PDEs. And asking about specific equations can sometimes bring out illustrations of general techniques. (This is not just for PDEs, but also for group theory [applications of Sylow theorems], number theory [applications of Chinese remainder; quad. reciprocity], and practically all other fields of mathematics.) So closing a question as too localised because it is about a specific equation will just be absurd.
- Depends. Like Mariano says: MO is for research level questions. Some subjects are inherently better suited for MO by virtue of it being at the forefront of research (topos theory etc.), compared to subjects that have been so well studied that it is in the common curricula (single variable calculus). But for most subjects the distinction is whether the question arose from research or graduate (postgraduate in Brit-speak) activities.
Now, in regards to (3) above, the more theoretical aspects of PDEs are often only studied in graduate schools in North America. Very few undergraduate programs (that I know of) offer PDE courses beyond the method of characteristics for first order scalar equations and series/fundamental solutions of constant coefficient linear PDEs. So that sets the bar a bit high in the get-go.
Furthermore, the technical nature of the field means that researchers basically know everyone whose work is related to theirs, so, for example, in my case, if I have a question about a particular PDE that I am interested in, more often than not I will just fire off an e-mail to the two or three people I know who would have a chance at knowing the answer, instead of bothering to ask them on MO or MSE.