First of all I would like to construct a semi formal sentence, such that the universum has at least three elements. My attempt:
$$\exists x\exists y\exists z (x\not=y\wedge y\not=z\wedge x\not=z)$$
Secondly, is there a (possibly infinite) set of sentences $T$ which has the infinite structures as models? I think it has something to do with Tarskis definition of truth. I use the following notation: $M\vDash \phi$ means $M$ satifies $\phi$, i.e the sentence $\phi$ is valid in $M$
Thirdly, how would you argue that there is no sentence $\phi$ which has the finite structures as models, I mean without a concret proof?
You've correctly written a sentence which is true only if the universe has at least three elements.
Now consider the set of all sentences 'There is at least one thing', 'There are at least two things', 'There are at least three things', 'There are at least four things", etc. etc. How big would a model for all these sentences taken together have to be?
The compactness theorem says that if $\Sigma$ is an inconsistent set of sentences, then there's a finite subset of $\Sigma$ which is inconsistent. This implies [the proof is given in the Wikipedia entry] that if $\varphi$ has arbitrary large finite models, it has an infinite model. So there can't be a $\varphi$ which has all finite structures (however large) as models but no infinite models. So what you need to understand here is compactness ...