Unsure how interpret the results of Spearman’s rank. My p-value is <0.05, but $\rho$ is close to zero so no correlation? data:  conflict$Livestock_density and conflict$Forest_density
S = 1e+08, p-value = 0.04
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho 
-0.0694 

Does $\hat{\rho} = -0.0694$ mean I should interpret test as not providing evidence of association, even though I  have a low p-value?

I’m guessing you have a huge sample. When this happens, your test gets very sensitive, so sensitive that it can detect tiny deviations from the null hypothesis.
Consequently, you have evidence of nonzero correlation, and that nonzero correlation seems to be quite small in magnitude.
Being slightly different from the null value is one way that the null hypothesis can be false.
