Whether can we infer that $X_1$ and $X_2$ are independent if we know they are conditional independence given $D$ and $\overline{D}$？ If we assume that:
(1) $P(X_1, X_2|D)=P(X_1|D)P(X_2|D)$, i.e., $X_1$ and $X_2$ are conditional independence given $D$;
(2) $P(X_1, X_2|\overline{D})=P(X_1|\overline{D})P(X_2|\overline{D})$, i.e., $X_1$ and $X_2$ are conditional independence given $\overline{D}$.
(PS: we cannot derive (2) with the assumption of (1))
Can we infer that $P(X_1,X_2)=P(X_1)P(X_2)$?

No.
Here are confusion matrices for a counter-example:
given $D$:
[[1,2], [2,4]]
given $D'$:
[[6,3], [2,1]]
You can see that the odds of $X_1$ are 1:2 if $D$ and 2:1 if $D'$ independent of $X_2$.  If you add these two, though, you get:
[[7,5], [4,5]]
