I'd suggest first rephrasing it in English as a sentence that uses more and more of the formal logic expressions: something like
"For any towers x and y, x and y have the same color"
(Here I'm taking advantage of the fact that it is equivalent to say that a whole class of elements shares a property and to say that every pair of elements in that class shares the property).
then we can further rephrase this as
"For any towers x and y, x has color c if and only if y has color c"
and finally
"For any towers x and y, any c is the color of x if and only if it is the color of y"
Then we need to write that symbolically, remembering that quantifying over certain types of things (towers in our case) can be written as quantification over an implication:
"For all x and y, if x and y are towers, then any c is the color of x if and only if it is the color of y"
$\forall x \forall y \left ( \left ( \operatorname{Tower}(x) \& \operatorname{Tower}(y) \right ) \longrightarrow \forall c \left ( \operatorname{Color}(x,c)\leftrightarrow \operatorname{Color}(y,c) \right ) \right )$
If you want, you can pull the quantification over the $c$ variable out front:
$\forall x \forall y \forall c \left ( \left ( \operatorname{Tower}(x)\& \operatorname{Tower}(y) \right ) \longrightarrow \left ( \operatorname{Color}(x,c)\leftrightarrow \operatorname{Color}(y,c)\right ) \right )$
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.
"There is an author who has not written a book".
$\iff$ "There exists an $a$ such that $a$ is an author AND, for all $b$, IF $b$ is a book, THEN it is NOT the case that book $b$ was written by author $a$."
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.
Best Answer
I don't know what your underlying language is, but if your set of variables includes $X,Y,x$ and your signature is just consisting of $\in$, taking $\subseteq$ as exactly the abbreviation as the right hand side, then (i) is a well-formed formula in the language of set theory. You may even add $\subseteq$ as a binary relation symbol to make it well-formed in some other signature of your taste.
(ii) seems to suggest that $P$ is a unary predicate which is quantified at the beginning, thus this formula seems not be well-formed in first-order logic.
If you assume for (iii) that $t$ is a variable symbol, and $G,U$ are unary/binary predicate symbols respectively and $P$ is a unary predicate, then this formula does not really make sense, as we have a predicate applied to another predicate.
I think there is needed information missing to really answer the questions for which you should search yourself(in the book where this question is from). The question
is always to be read with some appropriate context.
Now, what is the set of well-formed formulas, $FO(\sigma)$? This set is generated in the following way from the before mentioned sets:
First, we form the set of terms of this signature $\sigma$.
Then $FO(\sigma)$ is formed using this set of terms and $\sigma$ again:
We remark a few things on the process of forming the set of well-formed first-order logic formulas over a signature $\sigma$ and a set of variable $Var$ I just sketched:
So for you to analyse if a formula is well-formed, your first question has to be over what signature and over what set of variables. Then you may assert membership in $FO(\sigma)$.
As I said before, this is just a sketch. First, what I was describing was first-order logic without equality, but the differences are marginal(on the syntactical level).
We all use shorthand notations and deviate from this formal specification. Thus e.g. even when some author formally defined $Var=\{x_1,x_2,\dots\}$, you're likely to see $\forall x\phi$ as a formula. This is meant to be a shorthand but it is not technically fine, but such a minor detail that normally no one gets hurt. A good mantra is that you are allowed to deviate if you can always rephrase it properly.
Another common shorthand also used in your example is that binary predicates are often denoted in infix notation, that is, instead of writing $>(x,y)$ you write $x>y$.
There are however situations where the concrete syntactical structure matters, that is why this is set on such a firm ground.
I want to end with, that working with these objects is a purely syntactical matter. There is no meaning attached to these strings(yet) and that is how they ought to be treated. Two strings differ if they differ in one symbol already. It is not about representing (semantically) the same.
EDIT: I tried to not give you an answer but to explain to you the context of how to find an answer yourself(which some of my perspective on the top). I could not have given a precise answer as you we're asking about a syntactic technicality for which I don't know the complete surrounding to, i.e. the signature, etc. This answer turned out longer than expected and thus if something is unclear, we should clarify this together in the comments.