  Classical novae are the result of thermonuclear flashes of hydrogen accreted
by CO or ONe white dwarfs, leading eventually to the dynamic ejection of the
surface layers. These are observationally known to be enriched in heavy
elements, such as C, O and Ne that must originate in layers below the H-flash
convection zone. Building on our previous work, we now present stellar
evolution simulations of ONe novae and provide a comprehensive comparison of
our models with published ones. Some of our models include exponential
convective boundary mixing to account for the observed enrichment of the nova
ejecta even when accreted material has a solar abundance distribution. Our
models produce maximum temperature evolution profiles and nucleosynthesis
yields in good agreement with models that generate enriched ejecta by assuming
that the accreted material was pre-mixed. We confirm for ONe novae the result
we reported previously, i.e.\ we found that $^3$He could be produced {\it in
situ} in solar-composition envelopes accreted with slow rates ($\dot{M} <
10^{-10}\,M_\odot/\mbox{yr}$) by cold ($T_{\rm WD} < 10^7$ K) CO WDs, and that
convection was triggered by $^3$He burning before the nova outburst in that
case. In addition, we now find that the interplay between the $^3$He production
and destruction in the solar-composition envelope accreted with an intermediate
rate, e.g.\ $\dot{M} = 10^{-10}\,M_\odot/\mbox{yr}$, by the $1.15\,M_\odot$ ONe
WD with a relatively high initial central temperature, e.g.\ $T_{\rm WD} =
15\times 10^6$ K, leads to the formation of a thick radiative buffer zone that
separates the bottom of the convective envelope from the WD surface. (Abridged)
