Re: Exactly what is 'waving' at 90 degrees to an accelerating charge?

Well, with the example of 60Co beta decay, I have several comments.  The 
beta decay, the result of the weak nuclear force, creates 60Ni in an 
excited state.  The subsequent emission of gamma rays is a separate 
process, the decay of this excited state to a state of lower energy/  
Actually two decays, each emitting one gamma ray whose energy comes from 
the reconfiguration of the wavefunction of the protons...essentially from 
redistributing their spatial locations so that the interaction between 
them is stronger.  These decays are the result of the strong nuclear 
force (quite different from the weak force) accelerating protons, not any 
recoil from the emitted electron.  In the quantum world, which you must 
deal with to truly comprehend nuclear interactions and the emission of 
photons, such classical concepts as a stirred up soup of protons caused 
by the recoil of an energetic electron (not very energetic, compared to 
the binding energy of the nucleus, by the way) are ill-defined and 
imprecise.  More mathematical descriptions are better-formed.  The 
rearrangement of the protons in the nucleus does represent a spatial 
rearrangement of charges which causes the emission of the gamma rays.  A 
more precise description really requires a knowledge of quantum mechanics.

Further, EM waves are not necessarily emitted precisely at right angles 
to accelerated charges.  It's true that the distributions of emitted 
photons tend to peak at such angles, but there's a spread of angles.  The 
electrons in the sun are low-energy, and the spread is quite broad (it 
gets sharper at higher energies due to relativistic effects).  You are 
correct that photons from the sun are coming from accelerated charged 
particles.  The photons emitted from the sun are emitted by accelerating 
electrons (not protons, they don't interact with the photons so much) in 
the photosphere of the sun, which is a hot dense plasma in a strong 
magnetic field.  It emits photons very much like a simple blackbody, 
governed by quantum mechanics you can look up easily in the beginning of 
any college thermal physics book.  Kittel and Kroemer wrote quite a nice 
book on thermal physics, which explains such things in detail and 
provides all the mathematical backing you need (it will take some time) 
to cover the subject of blackbody radiation.

As far as the radiation propogating through space, the picture of an EM 
wave is somewhat simple.  You have a time-varying electric field, changes 
in which cause a time-varying magnetic field.  The variation in the 
magnetic field causes a variation in the electric field.  It's like a 
spring going back and forth, trading energy between the two fields.  As 
far as a deeper description of what such fields consist of at their most 
basic levels, your final descriptions seems to begin to scratch this 
surface...but the answers lie deep in string theory which is still being 
worked on.

You seem to be thinking very deeply on the subjects of the emission of 
gamma rays.  I suggest starting with quantum mechanics and atoms before 
going to considering nuclear decays.  The issues in that situation are 
more clear-cut, and provide a better introduction to quantum mechanics.  
Griffiths wrote both an excellent EM book and an excellent introduction 
to quantum mechanics.  They're usually used as textbooks for college 
students, but his descriptions are very clear to put the concepts in mind 
for general understanding.  I reccomend you start with the books I 
mentioned, the comprehensive picture you seek has many different aspects 
and cannot be found on a simple website...even one as informative as the 
madsci network.  :)