Precisely, is $k(x_i- x_j) = -\|x_i-x_j\| \quad x_i, x_j \in \mathbb R$ a valid kernel?
I know that the absolute value of kernel formulation is not a valid kernel since it is not positive semi-definite. However, the above-mentioned kernel in the literature is referred to as an energy kernel and it is used to calculate energy distance.
https://en.wikipedia.org/wiki/Energy_distance
https://pages.stat.wisc.edu/~wahba/stat860public/pdf4/Energy/EnergyDistance10.1002-wics.1375.pdf
In brief, is it a kernel ($k$), if not why it is referred to as an energy kernel?
Different authors use the term "kernel" differently. In the machine learning community, some similarity measures are called kernels, such as $$k(x,y) = \begin{cases} 1 , \text{ if } \|x-y\| \le \epsilon \\ 0 , \text{ else} \end{cases}$$ that is popular in spectral graph clustering. However, by choosing a data set $\cal X = \{0,1,2\}$ and $ \epsilon = 1$, you end up with a kernel matrix that has negative eigenvalues, so this kernel is not PSD.
On the other hand, mathematics tend to use kernels as a synonym for PSD kernels. Mercer's theorem guarantees that then, and only then, does there exist a Hilbert space $\cal H$ in which the inner product is described by $k$, in the sense that there exist map $\phi$ such that $$\forall x,y \in \cal X: \langle \phi(x) , \phi(y) \rangle_\cal H = k(x,y)$$
Now to your kernel: It is not PSD. Take $\cal X = \{0,1\}$ and check the resulting kernel matrix.