I think the statement you asked question 3 about describes Vallado, Kelso, and associates, nearly all of whom are former USAF Space Command people with experience inside the unit that uses SGP4 operationally, carefully comparing results from their reconstructed version of the code to results from the official code, in search of places they need to change their code to better match what's being done inside the latest SGP4. If they didn't see any sign of it, then I doubt anyone else would, either, so I can't help you on question 4.
You asked earlier whether SGP4 really changes anymore, and I have one thing to add on that topic now: yesterday I downloaded SGP4 code version 8.2, released just a couple of weeks ago (the link was broken for a while, but it's back in working order now). The release notes say they "Fixed some bugs hindering performance. SGP4 v8.2 is now over twice as fast as v8.1 (over four times as fast as v8.0)", so it appears they are indeed continuing to modify the official SGP4 code, even if only slightly.
Question 2, however, I can now answer. The reference you need is Hoots 1980, A Short Efficient Analytical Satellite Theory, Journal of Guidance, Control, and Dynamics 5.2 (194-199). This is the article I mentioned in this answer as being paywalled, although now that I have my hands on a copy, I note that this particular article states explicitly, "This paper is declared a work of the U.S. Government and therefore is in the public domain", so perhaps a complaint to the publisher (AIAA) is in order.
In the spirit of that statement about the public domain, let me quote some of the historical commentary:
One of the first theories of this type was developed by Hilton and
Kuhlman <1> in 1966. (For a more recent reference see Hoots and Roehrich. <2>)
This simplified general perturbations theory,
hereafter called SGP, uses a simplification of the work of Kozai <3> for
its gravitational model and it takes the drag effect on mean motion as
linear in time. This assumption dictates a quadratic variation of mean
anomaly with time. The drag effect on eccentricity is modeled in such
a way that perigee height remains constant.
In 1970 Cranford (see Lane and Hoots <4>) developed a simplified general perturbations theory
called SGP4. This model was obtained by simplification of the more
extensive analytical theory of Lane and Cranford <5>, which uses the
solution of Brouwer <6> for its gravitational model and a power density
function for its atmospheric model.
The SGP8 theory is obtained by simplification of an extensive
analytical theory of Hoots <7> which uses the same gravitational and
atmospheric models as Lane and Cranford did but integrates the
differential equations in a much different manner. The full theory is
valid for all eccentricities between 0 and 0.1 and all inclinations
not near 0 deg or critical inclination. Since the full theory
contains several terms which only become important for larger
eccentricities, it was felt many terms could be dropped without
affecting predictions on most satellites. Furthermore, it can be shown
that many terms included in the full theory are much smaller than the
differences introduced by using the power density function atmosphere
to model the real-world atmosphere. Thus, in an operational
environment, many of the differences between the full theory and the
simplified theory will be masked by differences between the model and
the real world.
The descriptions of the changes, and the justifications for them, are:
The first type of term to be considered for simplification was the
short-period drag periodics of Eqs. (3). It was found that these terms
could be excluded from the theory with negligible effect. Second, we
dropped terms of the form of drag coupled with drag which appear in
the secular Eqs. (1) for mean anomaly and argument of perigee. The
next type of term examined was of the form of drag coupled with
gravity. Although terms of this form occur in the triple-primed
differential equations for all six orbital elements, it was found
that the direct effect on each of the elements was very small.
However, since changes in the mean motion and eccentricity directly
cause changes in total drag and since the mean motion is integrated a
second time in the mean anomaly equation, it was found that the
dominant part of the coupled terms must be retained in the mean motion
and eccentricity differential equations, but all coupled terms can be
neglected in the other four differential equations.
It is well known
that the gravitational model of Vinti <9> allows algebraic combination of
the second and fourth zonal harmonic terms. Since the Vinti potential
differs from the true gravitational coefficients, we found that
adopting the Vinti relationship $J_4=-J_2^2$ caused measurable degradation.
However, if we use the Vinti potential in the long-period terms only
and retain the values of $J_2$ and $J_4$ in the secular terms, we found a
significant algebraic simplification in our equations with little loss
of accuracy. In addition, adopting the Vinti potential removes from
SGP8 any singularities at the critical inclination.
simplification concerned the purely gravitational terms. It was found
that terms of size second order times $e^2$ could be neglected in the
secular gravitational terms. Additionally, the gravitational periodics
can be simplified considerably. It has been shown by Hoots <10> that the
Lyddane-modified Brouwer geopotential transformation can be
reformulated in terms of an alternate set of variables which allows a
direct conversion from double-primed elements to Cartesian position
and velocity while reducing the formula amount of the transformation
by one-third. By using this alternate set of variables and retaining
only the dominant periodic terms, we obtained a significant decrease
in the number of terms in the transformation while sacrificing little
in accuracy for most satellites.
References to "double-primed" and "triple-primed" equations and elements assume you have copies of Brouwer <6> and Lyddane <8> at hand and are following along in their notation.
The citations from the paper are:
<1> Hilton, C.G. and Kuhlman, J.R., "Mathematical Models for the Space
Defense Center," Philco-Ford Corporation, Colorado Springs, Colo.,
U-3871, Nov. 1966.
<2> Hoots, F.R. and Roehrich, R.L., "Models for Propagation of NORAD
Element Sets," Aerospace Defense Command, Peterson AFB, Colo., Project
SPACETRACK Rept. No. 3, Dec. 1980.
<3> Kozai, Y., "The Motion of a Close Earth Satellite," Astronomical
Journal Vol. 64, Nov. 1959, pp. 367-377.
<4> Lane, M.H. and Hoots, F.R., "General Perturbations Theories
Derived from the 1965 Lane Drag Theory," Aerospace Defense Command,
Peterson AFB, Colo., Project SPACETRACK Rept. No. 2, Dec. 1979.
<5> Lane, M.H. and Cranford, K.H., "An Improved Analytical Drag Theory
for the Artificial Satellite Problem," Paper 69-925 presented at the
AIAA/AAS Astrodynamics Conference, Aug. 20-22,1969.
<6> Brouwer, D., "Solution of the Problem of Artificial Satellite
Theory Without Drag," Astronautical Journal, Vol. 64, Nov. 1959, pp.
<7> Hoots, F.R., "Theory of the Motion of an Artificial Earth
Satellite," Celestial Mechanics, Vol. 23, April 1981, pp. 307-363.
<8> Lyddane, R.H., "Small Eccentricities or Inclinations in the
Brouwer Theory of the Artificial Satellite," Astronautical Journal,
Vol. 68, Oct. 1963, pp. 555-558.
<9> Vinti, J.P., ''New Method of Solution for Unretarded Satellite
Orbits," Journal of Research of the National Bureau of Standards,
Vol.62B, 1959, pp. 105-116.
<10> Hoots, F.R., "Reformulation of the Brouwer Geopotential Theory
for Improved Computational Efficiency," Celestial Mechanics, Vol. 24,
Aug. 1981, pp. 367-375.
<11> Lerch, F.J. et al., "Gravity Model Improvement Using GEOS-3 (GEM
9&10)," Goddard Space Flight Center, Greenbelt, Md., GSFC Report No.
X-921-77-246, Sept. 1977.
<12> Jacchia, L.G., "New Static Models of the Thermosphere and
Exosphere with Empirical Temperature Profiles," Smithsonian
Astrophysical Observatory, Boston, Mass., SP-313, May 1970.