1) There are at least two people who everyone knows. Domain = {People}
My take in this.... for the 1) part is it valid to do something like this ∃𝒙∃𝒚∀z𝑷(𝒙, 𝒚,z). Where in my words i could be totally wrong.. there exist a pair (x,y) who everyone (z) knows.
You must say: "There are some $x$ and some $y$ who are not the same people and every $z$ will know $x$ and know $y$."
You should also use a bivariate predicate such as $\def\op#1{\operatorname{\rm #1}}\op{K}(~,~)$ for "_ knows _"
$$\exists x~\exists y~\forall z~\bigl(x\neq y\wedge \op{K}(z,x)\wedge \op{K}(z,y)\bigr)$$
2) Every student takes at least two classes. Domain = {people, classes}
"For every $x$ who is a student, then there is an $y$ which is a class that is taken by $x$ and there is a $z$ which is another class that is taken by $x$."
You will need predicates for: $\op P(~)$ "_ is a people", $\op C(~)$ "_ is a class", and $\op T(~,~)$ "_ takes _" .
3) All Students know each other. Domain ={ All people}
"For every $x$ who is a student, then for every $y$ who is a student, then $x$ knows $y$."
Use predicates $\op S(~)$ for "_ is a student", and $\op{K}(~,~)$ for "_ knows _".
(NB: do you need to worry about whether $x$ and $y$ are the same people?)
a) “Every state has exactly one head of state.”
my answer: ∀x (Sx → ∃1y Hyx)
There are several ways to formalise the given sentence; one is $$∀x \,\big(Sx → ∃p ∀q \,(Hqx ↔ q=p)\big).$$
b) “Batman and nobody else but Batman can save the world.
b: Batman.
My answer:∃b (Sb & ∃y (Sy & y = ¬ b) )
Since you have defined $b$ as a constant (rather than a variable), it doesn't make sense to write “$∃b$”.
Part (b) is strictly easier than part (a), so use the structure of my suggested formalisation above to figure out this answer. Note that here we don't need “$∃p$” since we are referring to a specific object $b.$
c) “There are at least two concrete objects.”
My answer: ∃x ∃y (Cx & Cy)
This is actually equivalent to $∃x \,Cx,$ since $x$ and $y$ can point to the same object.
Hint: try $$∀x∃p\,\big(Cp\ldots\big).$$
Best Answer
You are a bit off on your answers, and one of the reasons is because you haven't delineated the scope of some of the quantified variables; as a result the reappearance of such a variable outside of the scope of its quantifier is then free.
Another more substantive problem is it seems you haven't grasped the general form for a universally quantified statement versus an existentially quantified statement.
For example: "All humans are mortal": This is a universally quantified statement. If we let $H(x)$ represent "x is a human," and let $M(x)$ represent "x is mortal", then what we are essentially saying, in loglish, is "For all x, IF x is human, THEN x is mortal". This translates, symbolically, to the following:
$$\forall x\,(H(x) \rightarrow M(x))\tag{1}$$
Compare the above to the following: suppose we had written:
$$\forall x\, (H(x) \land M(x))\tag{(2) incorrect}$$
What this incorrect translation says is: "For all x, x is human and x is mortal." This states that everything is human and everything is mortal, whereas what we want to say is something regarding all and only those things that are human.
With that in mind, try to rewrite your first statement accordingly.
On the other hand, the general form for an existentially quantified statement uses conjunction to assert "there exists something such that that something is P and that something is Q."
For example, suppose we want to translate: "Some student missed class today." Crudely, we can denote by $S(x)$: "x is a student." And we can denote by $M(x)$: "x missed class today." Then the symbolic translation amounts to:
$$\exists x\, (S(x) \land M(x)).$$
I'll deal with the second statement, in part to make explicit the scope of each quantified variable, and in part to correct the translation for the statement that includes both an existential and universal quantifier.
A full symbolic translation, then, gives us: $$\exists a \Big(a \in \text{ Author }\land \forall b(b\in \;\text{Book}\;\rightarrow \lnot \operatorname{by}(a, b))\Big)$$
Note that we want $\lnot$by$(a, b)$ since we are talking about book $b$ not being written by author $a$, per your definition.