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Also, it fails if you can find a vector bundle $E$ such that $H^1(X, E)\neq 0$ but $H^1(U, E|_U) = 0$ and $H^1(Z, E|_Z)=0$. This happens probably quite often, but I don't know if such examples exist in general.
Since sheaves on $X$ are a full subcategory of $D^b(X)$, if this is a monomorphism, then it is still a monomorphism in the category of sheaves on $X$. This I think may fail if $X$ has an embedded point, e.g. $X = \mathrm{Spec } k[x,y]/(xy, y^2)$ and $Z = \{0\}$ with reduced closed subscheme structure.
Isn't it true that $j^*$, $i^*$, $a^*$ and $j_*$ are exact? Also $a^* F$ is flat, so $i^* a^* F$ is just the usual (non-derived) pull-back? Also, $F$ is a direct sum of its cohomology. That is, you are basically asking if $\mathcal{O}_X\to \mathcal{O}_Z\oplus\mathcal{O}_U$ is a monomorphism. Is that right?
The short exact sequence $0\to W\to W\to W_a\to 0$, where the map $W\to W$ is multiplication by $p^a$, is a projective resolution of $W_a$ over $W$. Applying $\mathrm{Hom}(-, W_b)$ to this resolution, we get that $\mathrm{Ext}_W(W_a, W_b)$ is just $W_b/p^a W_b$.
See Hartshorne, proof of Theorem III 7.11 - there he computes ${\scr E}xt^r(\mathcal{O}_{Y_1}, \omega_X)$ by applying $Hom(-, \Omega_X)$ to the Koszul complex of $\mathcal{O}_{Y_1}$. The Koszul complex exists only locally, but then he shows that these local calculations glue together. I think you can apply the same argument to $\mathcal{O}_{Y_2}$ in place of $\omega_X$.
