I'll give brief statements with lots of hand-waving since I'm low on time. These are long questions.
1. In short, this is a deeply philosophical question. My personal inclination follows yours that the incompleteness theorems are overhyped. I don't buy Penrose's argument. Like most other formal limiting results philosophically I view those results as representing fundamental limits of both human and machine reasoning. While a single unproveable but "obviously true" result generally point to an inadequacy of the formal system, if every formal system suffers from some inadequacy that is indicative of a limitation in the fundamental human faculties of comprehension rather than a limitation of formal systems per se (the assumption e.g. that our formal systems must use finitistic methods is one born of human limitations and, depending on one's thoughts about the universe, potentially fundamental physical ones).
There are philosophically defensible positions that try to claim the incompleteness theorems have implications for truth in the real world, but I go back and forth on whether I believe them and more to the point they are far more subtle than the usual pop culture presentations of Godel's incompleteness theorems.
2. The mathematical Platonists I've come across believe that the continuum hypothesis is in fact false. (As an aside it's interesting that you find the axiom of determinacy natural as it contradicts the axiom of choice.) This mainly hinges on your opinion of the reality of large cardinals (which perhaps count as your "interesting" constructs), which many Platonists for a variety of reasons believe to be real.
1. In short, this is a deeply philosophical question. My personal inclination follows yours that the incompleteness theorems are overhyped. I don't buy Penrose's argument. Like most other formal limiting results philosophically I view those results as representing fundamental limits of both human and machine reasoning. While a single unproveable but "obviously true" result generally point to an inadequacy of the formal system, if every formal system suffers from some inadequacy that is indicative of a limitation in the fundamental human faculties of comprehension rather than a limitation of formal systems per se (the assumption e.g. that our formal systems must use finitistic methods is one born of human limitations and, depending on one's thoughts about the universe, potentially fundamental physical ones).
There are philosophically defensible positions that try to claim the incompleteness theorems have implications for truth in the real world, but I go back and forth on whether I believe them and more to the point they are far more subtle than the usual pop culture presentations of Godel's incompleteness theorems.
2. The mathematical Platonists I've come across believe that the continuum hypothesis is in fact false. (As an aside it's interesting that you find the axiom of determinacy natural as it contradicts the axiom of choice.) This mainly hinges on your opinion of the reality of large cardinals (which perhaps count as your "interesting" constructs), which many Platonists for a variety of reasons believe to be real.