About Question 1), basically, the formula :
$ \{$ odd numbers $\} = \{ n \in \mathbb{N} : \exists \, k \in \mathbb{N} \; (n = 2k+1 ) \; \}$
is a shorthand for the formal :
$\{ n : n \in \mathbb{N} \quad \land \quad \exists \, k \; ( k \in \mathbb{N} \land n = 2k+1 ) \;\}$.
The syntax of first-order set-theory, requires $\{ x : \phi(x) \}$
where $\phi(x)$ is a well-formed formula with one free variable (a well-formed formula is an expression built up according to "language specifications").
Due to the fact that $\in$ is part of set-theoretic language, you can use it in $\phi$, so that you can have :
$\{ x : x \in S \land P(x) \}$.
Accordingly, I'll rewrite : $\{ f(x) : x \in S \}$ as :
$\{ \; y : \exists \, x \, \exists \, z \quad (Funct(z) \quad \land \quad x \in S \quad \land \quad <x,y> \in z) \; \}$;
where $Funct(z)$ is a "complex" expression saying that the set $z$ is a function; again, the formula at the right of the colon has form : $\phi(y)$, with $y$ free.
Of course, in addition to the syntactical aspects, that dictates how to build well-formed formulas, we have the aspects related to "existence" of sets, that depends on the axioms.
In Axiomatic set theory (the form due to Zermelo and Fraenkel, called $\mathsf {ZF}$) you must use Axiom Schema of Separation in order to prove that the previous set exists.
This answer also the question about $\{ x \in S: P(x) \} = \{ ? : x \in S \}$. It must be :
$\{ x \in S: P(x) \} = \{ x : x \in S \land P(x) \}$.
Left of the colon we must have a set variable; right of the column, a "condition" specifying that the resulting set will be the subset of $S$ made by the elements $x$ of $S$ such that $P(x)$ holds.
It must be also a formula with constant symbol (a "name", like : $\emptyset$): for example, we may have :
$\{ x : x \in S \land x \in \emptyset \}$.
In this case we "choose" the $x \in S$ that in addition belongs to $\emptyset$; but there are none, so the result will be simply the empty set.
NOTE about language. In order to understand the formulas above, we need some preliminary notions about first-order language.
We start with symbols : variables: $x$, $y$, ..., predicates : $P$, $Q$, ..., connectives : $\lnot$, $\land$, $\lor$, $\rightarrow$ and the quantifiers $\forall$ and $\exists$. We may add also constants, like $0$ and $1$ in arithmetic and $\emptyset$ in set theory.
A "special" (binary) predicate is $=$ (both in arithmetic and set theory), while the (binary) predicate $\in$ is used in set theory.
They are written usually - mainly due to tradition - in the infix form, i.e. $x \in y$ and $x = y$, instead of the "official" prefix form, i.e. $\in (x,y)$ and $=(x,y)$.
Infix form is more readible for humans; computers "prefer" the prefix one.
Variables and constants are terms : they behave like "nouns".
With predicates and connectives and quantifiers you can build formulas, like $x \in \emptyset$, $0 = 1$.
Roughly speaking, terms have denotation and formula have meaning: but, in order to achieve "meaningfulness", we must follow the rules of formation (the syntax of the language, like the formal specifications for a programming language).
The matter is like in natural language : the phrases "the flower is red" and "the man run slowly" are "well formed", while the phrase "the man runs redly" is meaningless.
In set theory we have that formulas like : $x \in A$ and $A \subseteq B \cap \emptyset$ are well-formed, i.e. they have meaning.
The expression $x \in \lor A$ is ill-formed, i.e. meaningless.
With quantifiers and connectives you can "build up" complex formulas (starting from "atomic" or elementary ones).
Examples from set theory.
Set theory add to the "basic" symbols (variables, connectives, quantifiers and equality ($=$)) only one predicate (bynary : $\in$) as primitive : all other symbols "specific" of set theory will be defined.
Please note: also the "name" for the empty set ($\emptyset$) is defined; it is introduced after we have proved that, according to the axioms of our theory, there exists a set that has no members, and that this set is unique.
Atomic formulas : $x \in y$, $x = y$, etc.
From this "austere" groundfloor we can buil all we need, i.e. complex formulas like : $x \in y \land x = y$, $\lnot x \in y$ (abbreviated as : $x \notin y$), ...
When we write a formula like $\phi(x)$, we usually want to refer to an (atomic or) complex formula with one free variable, like $x \in \emptyset$. But we can also write : $x \in \emptyset \land x \notin x$. This last formula has the "form" $x \in S \land P(x)$ (where "incidentally" $P(x)$ is the predicate of the Russell's paradox).
Our first examples are of this "form": the set of even numbers is the set of all $x$ such that $x \in \mathbb{N} \land \exists y (x=2 \times y)$; here $\exists y (x=2 \times y)$ is a formula with the free variable $x$, like $P(x)$.
Note we have implicitly "added" to set language also symbols for arithmetic, like :$\mathbb{N}$, $+$, $0$, $\times$. Please, assume for the sake of discussion that it is admissible.
NOTE on functions in set theory.
Functions in set theory are a particular type of sets (in set theory - "obviously" - all is a set).
We nedd the concept of ordered couple $(a,b)$ that is different from $\{ a, b \}$ (because $\{ a, b \} = \{ b, a \}$, i.e. the order is immaterial, where the ordered couple is ... ordered); $a$ is the first element of the couple and $b$ is the second.
A function in set theory is a set of ordered couples,
provided that it satisfy the "basic rule" for functions, i.e. that if $f(x)=y_1$ and $f(x) = y_2$, then $y_1=y_2$.
This rule will be "rewrited" in set language as : a set $f$ of ordered couples is a function when :
$\forall x \forall y_1 \forall y_2 ( \quad <x,y_1> \in f \quad \land <x,y_2> \in f \quad \rightarrow y_1 = y_2 \quad )$.
We will write the function $f : \mathbb{N} \rightarrow \mathbb{N}$ defined as $f(x) = 2 \times x$ as :
$\{ \quad <x,y> \quad : \quad x,y \in \mathbb{N} \land y = 2 \times x \quad \}$.
This "object" in set theory (it is a set !) behaves like "usual" mathematical functions.
