It appears that you and your colleagues have a few misguided ideas from whatever sources ... The question (I don't have the practice questions so I am running with your restatement of the question) is posing the situation of an inversion in the general air mass which, in general, puts a lid on vertical movement - consider the typical inversion situation over an industrial area .. all the pollution sits below the inversion level and, above, it is nice and clear. In the situation posed by the question, it is expected that a parcel of air will cool while rising. Depending on the starting temperature and the temperature of the air mass relative to the rising parcel, the densities generally will be such that the parcel's tendency to continue rising decreases and then reverses. Eventually, it finds an equilibrium density level and is quite content to sit there ... What the inversion does is to make the change from "continue rising" to occur somewhat more quickly than normal. With an inversion, the normal temperature reduction with increasing height reverses for a small height band, giving a bit of a blip to the "standard" variation plotted line in the link.
Our understanding is that temperature is increasing with height, as the air is rising it gets warmer making it less dense than the environment.
It appears you might be confusing the air mass environment's temperature profile and the variation in the temperature of a rising air parcel. For the air mass, the temperature variation depends on the level considered for lapse rate. For pilot training one presumes we are looking at the troposphere, where the usual pattern is for air temperature to decrease with increasing height. For instance, see
eesc.columbia.edu/courses/ees/slides/climate/atmprofile.gif. (Kelvin and Celsius are the same unit size but start at different zero points .. take 273 from K to get C). Considering a parcel of air, rising for some reason, the generally decreasing amount of atmospheric stuff above the parcel gives rise to a reducing density and pressure. This allows the rising parcel to expand and, generally, one expects to see a decrease in the temperature of the parcel of air as it expands. It is how the temperature and variation of temperature comparing the general air mass to the parcel of air progress which determine whether the situation is stable or unstable.
This gives way to an unstable environment.
It is the temperature of a parcel of air (rising for whatever reason) compared to that of the general air mass, which provides a basis for assessing stability or instability.
It will continue to rise because of the negative lapse rate.
You need to consider the relative temperatures (and densities) at a level, rather than the absolute lapse rate.
So some decided that the probability of vertical motion is high as the air is warmer than the environment.
Probably the reverse applies if you consider the expected temperatures of the general air mass and the rising parcel of air.
Other ideas are that As air begins to rise, it will cool adiabatic-ally and become cooler than the environment.
That's probably a lot closer to the real story ...
The books answer is" Low as rising air becomes cooler than its environment."
Which is what most of us would expect to see ..
