Here's an alternative construction that doesn't require moduli spaces. For simplicity, let's assume that the field $k$ is perfect, let $\bar k$ be an algebraic closure of $k$, and let $G_k=\mbox{Gal}(\bar k/k)$. For any subvariety $Y\subset X$ defined over $\bar k$, let $G(Y)=\{\sigma\in G_k : \sigma(Y)=Y\}$. Then the natural field to associate to $Y$ is $k_Y:=\bar k^{G(Y)}$, i.e., the fixed field of $G(Y)$. In particular, you can apply this construction to your $D_i$.

As for your last question, first you need to define $k_Y$ for subschemes that might not be irreducible or reduced, since your intersection $D_i\cap D_j$ is a union of subvarieties of codimension 2 (and it really should be treated as a subscheme, not a simple set theoretic intersection). But a similar Galois-theoretic construction should work, after which Will's inclusion $k_{D_i\cap D_j}\subseteq k_{D_i}k_{D_j}$ follows from basic Galois theory.

A final note. If, say, $D=D_1+D_2$, then $k_D$ may be strictly smaller than $k_{D_1}\cap k_{D_2}$. For example, taking divisors in $\mathbb{P}^1_{\mathbb Q}$, let $D_1=(i)$ and $D_2=(-i)$. Then $k_{D_1}= k_{D_2}=\mathbb{Q}(i)$, while $k_D=\mathbb{Q}$.