The domain of a first order structure contains all possible inputs to the functions of the first order structure. Does it also contain all possible outputs of the functions? (Answered in comments as yes)
Are valuations also elements of the domain? Are these two equivalent? I'm getting really confused about the notation for satisfiability.
Long comment
The "objects" belonging to the domain are the possible values for the variables. Thus, they are the possible "input values" for the functions.
The function symbols of the language are interpreted as function on the domain whose output values are themselves objects of the domain.
Consider the simple example of the language of arithmetic with $+$; it is interpreted as a binary function (sum) having a pair of numbers as input and a number as output $+ : \mathbb N \times \mathbb N \to \mathbb N$.
Valuations (aka: variable assignment functions) are function from language to domain of the interpretation that assign to each variable (and then to terms) an object of the domain. Thus, the "outputs" of valuations are objects of the domain.
Trying to mimic your symbols, the basic property of satisfaction that involves valuations is:
let $M$ a model, $A(x)$ a formula with free var $x$ and let $v$ a variable assignment such that $v(x)=d$ where $d$ is an element of the domain of $M$; we have that $M,v \vDash A(x) \text { iff } M \vDash A[x \leftarrow d]$.
