Decidability vs Completeness

Your summary seems accurate, with one exception: The theory of algebraically closed fields of characteristic 0 is complete. Perhaps you meant the theory of algebraically closed fields, without specifying the characteristic?

As Chris Eagle said, your example for (1) is wrong. Removing the characteristic specification does the trick (as they observe), but there are also far simpler examples. For instance, take the empty language $\{\}$ (so only "$=$" allowed, besides the pure logical grammar) and consider the theory $$T=\{\exists x,y\forall z(x=z\vee y=z)\}.$$ This theory has exactly two models up to isomorphism, a one-element set $M_1$ and a two-element set $M_2$. These aren't elementarily equivalent, so $T$ isn't complete, but it is decidable since we have $$T\vdash\varphi\quad\iff M_1\models\varphi\mbox{ and }M_2\models\varphi,$$ and checking whether a sentence holds in a finite structure is computable.