Say I have two signals, one being $g_1(t)$ and the other one being $g_2(t)$, and say that I am being asked to compute the energy of $g_3(t)$, where $$g_3(t) = g_1(t) + g_2(t)$$
Is it correct to compute the energies of $g_1(t)$ and $g_2(t)$ separately and simply add the two values to find $g_3(t)$ or is it correct to compute them as one value? I.e. $$\int_ {-T}^{T} \big[g_1(t)+g_2(t)\big]^2 \,dt $$
You simply have to apply the definition of energy. Assuming all signals involved are real, the energy of $g_3(t)$ is given by
\begin{align} E &= \int_{-\infty}^\infty g_3^2(t) \, dt \\ &= \int_{-\infty}^\infty \big(g_1(t) + g_2(t)\big)^2 \, dt \\ &= \int_{-\infty}^\infty g_1^2(t) \, dt + 2\int_{-\infty}^\infty g_1(t)g_2(t) \, dt + \int_{-\infty}^\infty g_2^2(t) \, dt. \end{align}
One interesting case is where $g_1(t)$ and $g_2(t)$ are orthogonal, in which case the middle integral is zero. In that case, the energy of $g_3(t)$ is the energy of $g_1(t)$ plus the energy of $g_2(t)$. One simple example is two rectangular pulses that do not overlap.
