Negative Heat Capacity of an Ideal Gas

I think your confusion is in understanding the definition of specific heat you have given. The idea is that we supply a known amount of energy as heat to our object and then look at the resulting temperature change. In other words, the specific heat tells us how much energy is needed to change the temperature of an object. You cannot apply this definition to any general process (it is not the specific heat of a process, it is the specific heat of the material).

In other words, just because a process involves transfer of energy as heat and a temperature change does not mean we just plug in those values into the specific heat definition and then say that is the heat capacity of that material. This is just not the case.

Yes, the specific heat capacity would be negative in that case. Of course it wouldn't be the heat capacity $c_V$ at constant Volume or $c_p$ at constant pressure. These are positive for ideal gases. It would be the heat capacity for some more unusual process, where the system gains more energy through work than it looses as heat (as you describe it in your question).

An example are polytropic processes, which obey

$$ p cdot V^n = mathrm{const.} $$

with a constant exponent $n$. The molar heat capacity for such a process is

$$ C_{mathrm{mol}, n} = R left[ frac{1}{gamma-1} - frac{1}{n-1} right], $$

where $gamma$ is the adiabatic index ($gamma = 5/3$ for a monoatomic ideal gas) and $R$ the gas constant. (See also this question about its derivation.)

If $1 < n < gamma$, the heat capacity $C_{mathrm{mol}, n}$ actually becomes negative.

I don't know where such a process is relevant, but you can make up a mechanical apparatus that realizes it, at least in theory. Maybe something like this:

The weight puts some pressure on the piston, but a heavy rope is attached to it, which is thicker in the middle. The rope runs over a pulley and if the piston goes up part of its weight pulls the piston up and reduces pressure. With the right mass distribution you should get $p propto V^{-n}$.

However, the negative heat capacity would mean that the system is unstable. If it comes in contact with a hotter reservoir, it absorbs heat, whereupon it expands and cools down so it can absorb even more heat and continue expanding. On the other hand, if the reservoir is cooler, it will continue to contract. I can imagine that, in other systems, this may ultimately lead to states which can be described by negative temperatures, but with an ideal gas this is not possible. Ideal gases can only have positive temperatures. Instead, the above contraption would eventually reach its limits, like the end of the piston or the end of the rope, where the pressure no longer follows a polytropic process.