  The current dynamical structure of the Kuiper belt was shaped by the orbital
evolution of the giant planets, especially Neptune, during the era following
planet formation, when the giant planets may have undergone planet-planet
scattering and/or planetesimal-driven migration. Numerical simulations of this
process, while reproducing many properties of the belt, fail to generate the
high inclinations and eccentricities observed for some objects while
maintaining the observed dynamically "cold" population. We present the first of
a three-part parameter study of how different dynamical histories of Neptune
sculpt the planetesimal disk. Here we identify which dynamical histories allow
an in situ planetesimal disk to remain dynamically cold, becoming today's cold
Kuiper belt population. We find that if Neptune undergoes a period of elevated
eccentricity and/or inclination, it secularly excites the eccentricities and
inclinations of the planetesimal disk. We demonstrate that there are several
well-defined regimes for this secular excitation, depending on the relative
timescales of Neptune's migration, the damping of Neptune's orbital inclination
and/or eccentricity, and the secular evolution of the planetesimals. We model
this secular excitation analytically in each regime, allowing for a thorough
exploration of parameter space. Neptune's eccentricity and inclination can
remain high for a limited amount of time without disrupting the cold classical
belt. In the regime of slow damping and slow migration, if Neptune is located
(for example) at 20 AU, then its eccentricity must stay below 0.18 and its
inclination below 6{\deg}.
