Sure, the problems the Ancients struggled over are now taught in high school, but I'm not sure who thinks we have unlocked most of the doors. I remind those people however that Wiler's proof of Fermat's Last Theorem was released only in 1994. Flipping through texts on group/ring theory (which has quite a few interesting applications actually), graph theory, and analysis (Riemann's Hypothesis is the big one here) suggests we still have a very long way to go.
I find your description of applied maths interesting: it's main thrust is for the general public? Given that, I would agree that everything that is needed for our (the general public's) lives have been found---thousands of years ago when our ancestors made the first calendars. I therefore disagree with your description, and say the handmaiden of applied mathematics is physics first and foremost. If we accept that, then there is still a very far way to go in understanding. Also, it would mean the development of fields like astrophysics drives mathematical research.
I was thinking about that jelly donut argument I made before; I think I may have been wrong. If you recall, back when all that revolutionary reworking in the calculus was going on (Newton, Liebniz, and just about everyone else got in on the action), proofs were often put on the backburner. Here we have an excited generation looking at strange possibilities, driven by naivete and even simple formalism to develop what we know today. The jelly must have been leaking out then, because the calculus was definitely a part of the enlightenment culture---it's no surprise that Liebniz' philosophy about 'monads' feels a lot like his work with infinitesimals in calculus.
It may seem unlikely that such a revolution in how mathematics is practiced may happen again, but I would keep it in mind. A lot depends, for example, on whether Riemann's Hypothesis is true; some mathematics today depends on it, and mathematicians have been a little impatient, deciding to go on. I definitely think Hilbert's list of problems is not the last word, and one day we may see another revolution of the same sort, where mathematicians "lose their heads" for a while.
Lastly, let's explore that idea about mathematicians searching for knowledge being its own reward, that, like you say, they <em>do</em> mathematics to pursue knowledge. Where did this drive come from? I'd guess from the Greek cultural tradition. What about now? The debates today seem to be: can a computer do mathematics---true mathematical research and proof? If the answer is yes, then culture in mathematics is either dead or an illusion. If not, though---if something would be missing from the programming, like the aesthetic quality, or the passion of pursuit---then human beings will always be integral in mathematical research. Aesthetic ideals are not immune to cultural influence.
Look at the jelly again, how proofs are structured. Not all proofs are created equal, meaning that some are more elegant than others. Not everyone liked the proof of the Four Color Theorem for example, calling it ugly. Look at intuitionism---if that took complete hold in mathematical practice then proof by contradiction would be disallowed and considered bad form. The jelly would change, being influenced by cultural ideas of what is considered a "good" proof.
Cheers
