  A major motivation of spectroscopic observations of giant exoplanets is to
unveil planet formation processes from atmospheric compositions. Several recent
studies suggested that atmospheric nitrogen, like carbon and oxygen, can
provide important constrains on planetary formation environments. Since
nitrogen chemistry can be far from thermochemical equilibrium in warm
atmospheres, we extensively investigate under what conditions, and with what
assumptions, the observable NH3 abundances can diagnose an atmosphere's bulk
nitrogen abundance. In the first paper of this series, we investigate
atmospheric T-P profiles across equilibrium temperature, surface gravity,
intrinsic temperature, atmospheric metallicity, and C/O ratio using a 1D
radiative-convective equilibrium model. Models with the same intrinsic
temperature and surface gravity coincide with a shared "universal" adiabat in
the deep atmosphere, across a wide equilibrium temperature range (250--1200 K),
which is not seen in hotter or cooler models. We explain this behavior in terms
of the classic "radiative zero solution" and then establish a semi-analytical
T-P profile of the deep atmospheres of warm exoplanets. This profile is then
used to predict vertically quenched NH3 abundances. At solar metallicity, our
results show that the quenched NH3 abundance only coincides with the bulk
nitrogen abundance (within 10%) at low intrinsic temperature, corresponding to
a planet with a sub-Jupiter mass (< 1 MJ) and old age (> 1 Gyr). If a planet
has a high metallicity ($\ge$ 10$\times$ solar) atmosphere, the quenched NH3
abundance significantly underestimates the bulk nitrogen abundance at almost
all planetary masses and ages. We suggest modeling and observational strategies
to improve the assessment of bulk nitrogen from NH3.
