Given a formula, phi, how do you go about finding an $L$-structure that satisfies it?

Question

The Question(s)

Let the language ${\mathcal L}$ be $\{S, <\}$, where $S$ is a unary function symbol and $<$ is a binary relation symbol.

Let $\phi$ be the formula:

$$(\forall x)(\exists y)(Sx < y)$$

1. Find an $L$-structure $\mathfrak A$ such that $\mathfrak A \models \phi$.
2. Find an $L$-structure $\mathfrak B$ such that $\mathfrak B \models (\lnot \phi$).
3. Prove your answer to either (1) or (2) is correct.
4. Write an $L$-sentence that is true in a structure $\mathfrak A$ if and only if the universe $A$ of $\mathfrak A$ consists of exactly two elements.

The Answer(s)

As I get used to structures, these are sort of answers that I have come up with, taking a lot of inspiration from answers such as this here. However, I was not able to approach the latter parts to come up with anything (I had ideas, but nothing formal). Would also appreciate any rectification on the parts that I have solved. Consider the following:

1. We are working in a language with one unary function symbol, $S$, and one binary relation symbol, $<$. So, to define a model $\mathfrak A$, all we need to do is specify a universe, a function $S^{\mathfrak A}$, and a relation $<^{\mathfrak A}$. Suppose that we let the universe $A$ be the set of $\mathbb N$ of natural numbers, and interpret the unary function $S^{\mathfrak A}$ as $S^{\mathfrak A}(x) = x + 1$ i.e. this is the successor function from $\mathcal{L_{NT}}$. Finally, we say that we will interpret $<^{\mathfrak A}$ for the "less than" relation, also similar to the one from $\mathcal{L_{NT}}$.

   We can see that $\mathfrak A \models \phi$ with the above specified $L$-structure.

2. Here, we need to go the other way, trying to specify an $L$-structure such that $\mathfrak B \models (\lnot \phi$). Suppose we let the universe be the collection of all finite strings of $0$ or more capital letters from the Roman alphabet. So, $B$ includes strings such as: $\epsilon$ (the empty string), BABY, COMEHEREAND, LEAVE, ORSTAY, CHRISTMASISNEAR etc.

   Interpret the unary function $S^{\mathfrak B}$ as the function that adds an X to the beginning of a string. So, $S^{\mathfrak B}(\epsilon)$ would be X, and $S^{\mathfrak B}$(SOHU) would become XSOHU etc.

   Finally, we say that we will interpret $<^{\mathfrak B}$ for the following relation:

   $<^{\mathfrak B} = \{(r, s) \in B^3 |$ r does not begin with an X $\}$

   We can see that $\mathfrak B \models (\lnot \phi)$ with the above specified $L$-structure.

3. Induction? Not sure how..?

4. Fix constants perhaps?