The answer to your second question is "no", but this doesn't have anything to do with $1$-dimensional asymptotic classes specifically, just ultraproducts.
Suppose $\mathcal{C}$ is a class of finite structures and $\mathcal{M}=\prod_{\mathcal{U}}C_i$ is an infinite ultraproduct of members of $\mathcal{C}$, where $\mathcal{U}$ is an ultrafilter on some index set $I$.
Now suppose $\phi$ is a first-order sentence true in $\mathcal{M}$. Let $J=\{i\in I:C_i\models\phi\}$. Then $J\in\mathcal{U}$ by Los's Theorem. It follows that the collection $\{C_i:i\in J\}$ contains structures of arbitrarily large cardinality. So, in particular, $\phi$ holds for infinitely many structures in $\mathcal{C}$.
To justify the last claim, suppose there is some integer $N$ such that if $i\in J$ then $C_i$ has cardinality at most $N$. Let $J'=\{i\in I:|C_i|\leq N\}$. Then $J\subseteq J'$, and so $J'\in\mathcal{U}$. By Los's Theorem (note that "cardinality at most $N$" can be written as a first-order sentence), $|\mathcal{M}|\leq N$, which contradicts the assumption that $\mathcal{M}$ is infinite.
EDIT: Someone else who studies 1-dimensional asymptotic classes more than me would provide a better answer to the first question. But a major motivation for the definition was to isolate broad combinatorial conditions on classes of finite structures under which behavior similar to the results of Chatzidakis, van den Dries, and Macintyre would be found. In general, when studying classes of finite structures through pseudofinite constructions, one always has ways of "counting" or "measuring" definable sets via the pseudofinite counting measure (which can be defined on sets in any dimension), for example. In one-dimensional asymptotic classes, this "counting and measuring" apparatus is much more finitely tuned, which allows for stronger results.
EDIT #2. The answer above shows that if $\mathcal{M}$ is an infinite ultraproduct of finite structures, and $\mathcal{M}\models\phi$, then $\phi$ is true in infinitely many members of $\mathcal{C}$. One can also arrange for $\phi$ to be false in infinitely many members of $\mathcal{C}$. For example, let $\mathcal{C}=\{G_n:n\geq 1\}$ where $G_n$ is the cyclic group of order $n$. (Incidentally, $\mathcal{C}$ is a $1$-dimensional asymptotic class.) Let $\mathcal{U}$ be a nonprincipal ultrafilter on $\mathbb{Z}^+$ which contains the set of even integers. Let $\mathcal{M}=\prod_{\mathcal{U}}C_n$. Then $\mathcal{M}$ satisfies the sentence $\phi$ saying "there is an element of order 2" since this sentence is true in $G_n$ for any even $n$. But there are infinitely many members of $\mathcal{C}$ where $\phi$ fails, namely $G_n$ for odd $n$.
