I want to know about:
- Statements which are tautological in all finite models but false in some infinite model.
- Statements which are tautological in all infinite models, but false in some finite model.
One example of 1 is "$G$ is a DAG implies $G$ has a sink". Another example of 1 is "there are an even number of odd degree vertices in $G$".
Here's an easy one for 2: "$\exists x\exists y(x\not=y)$." This is true in exactly those structures which have at least two elements, hence true in any infinite structure; at the same time, it is not true in a one-element structure.
For 1, you've already found some examples. My favorite, though, is a piece of hard algebra. The language here is the language of rings, and the sentence says $$(*)\quad\mbox{If this is a division ring, then it is commutative.}$$ (That is, it says [division ring axioms]$\implies$[commutativity]. It's tedious, but simple, to express it formally in first-order logic.) Clearly $(*)$ is false in some infinite structure (say, the quaternions); but surprisingly, it turns out that every finite division ring is commutative! This is a hard result - Wedderburn's little theorem. The nice proofs I know of each use group cohomology.