Definitions
from The Century Dictionary.
 noun An adiabatic curve or line.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun physics The curve on a graph of pressure and temperature which gives the adiabatic heating or cooling of a gas.
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

In El Nino the tropical atmosphere warms, and the vertical structure of this warming is indeed close to that obtained by assuming that the atmosphere stays near a moist adiabat, that is, the warming is substantially larger in the upper troposphere than near the surface.

Moist adiabats are much more complicated than the dry adiabat.

At typical Earthlike surface temperatures, the effect is very strong: for a starting temperature of 5ºC, the mean lapse rate over the first 5 km is 7.2 K km−1, for 20ºC it is 4.8ºC km−1 and for 35ºC it is 3.3ºC km−1 compare with 9.8ºC km−1 for the dry adiabat.

At high altitudes, especially where the moist adiabats approach the lapse rate of a dry adiabat, the temperature difference increases with temperature.

The radiating surface of Venus is at the cloud tops, as the pressure at the bottom of the atmosphere is 90 atm, the adiabat has a long way to go.

So if the atmosphere is warming more than the surface, this would not necessarily make it more stable depends on the slope of the temperature profile compared with the slope of the saturated adiabat.

Re question #2, the static stability of the tropics is really determined by the moist conditional instability associated with the moist saturated adiabat, the slope of which decreases as temperture increases.

Cooler tropical SSTs would reduce the absolute humidity of the lower troposphere and steepen the slope of the moist adiabat above the condensation level.

Also they say cooler SSTs would “steepen the slope of the moist adiabat…,” but that, too, is a little funky.

It is important to know the location of the weather station relative to the city because the dome of warm air that is the UHI adiabat is shifted by the wind.
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