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.

We can have undecidable and incomplete theories. e.g Peano Arithmetic

This is based on a very different definition of complete than what you wrote. Godel's Incompleteness Theorem uses the "if it is true then it is provable" pseudo definition of completeness. And he gets around the ambiguity of that definition by only needing to give 1 meaningful counterexample, a unary predicate $P$ with the quality that there is a proof for $P(0)$ and a proof for $P(1)$ and a proof for $P(2)$, etc, but there is no proof of $\forall x . P(x)$.

The definition of completeness you give is the one that a person would mean if they said "propositional logic is complete"; that is, that every propositional statement has a proof or disproof. But an IMO better way to phrase the definition in that case is "if it exists in this language, then it has a proof". In the definition there is no particular reason to separate cases according to $\lnot$.

If someone was to say a theory is complete, I'm not even sure I could guess what they mean. A theory is just a set of theorems (although usually in context, with some sort of deductive closure). It is usually meaningless to say a theory is (in)complete, except maybe relative to a grammar, you would instead say whether a logic is complete.

When they say "[a particular] first order logic" is complete, what they mean is that every statement that is a tautology (relative to whichever first order model theory they are using) has a proof in that logic. So when they talk about the completeness of [a particular] first order logic, in absolutely no way are they suggesting that it is decidable; that is, they are not at all alluding to the definition in the original question. It's all just first order model theory stuff.

Completeness is used to mean a lot of different things.