Relationship of lambda calculus to the rest of math I just started reading "The calculi of lambda conversion" by Church. 
Church defines functions like: id x = x, and says the domain and range are understood to be as permissible as possible. Permitting even itself, id id = id
In my experience, I've always been told to specify a domain and range with the functions I've defined. And they are usually relatively limited, in contrast to id.
This is the first time I've seen functions with a domain and range this large. Are there uses for functions with wide domains and ranges in mathematical contexts other then logic or lambda calculus?
 A: The question you are trying to ask is "What is a denotational semantics for the untyped lambda calculus?"
This is a difficult problem because, as Bjorn Kjos-Hanssen said in his answer, if you try and make variables range over elements of some set $D$ you find that you must have $D \times D \cong D$ and $D^D \cong D$. Unfortunately this implies that $D$ is the singleton set and all lambda terms must represent the same function.
Dana Scott solved the problem of giving a denotational semantics to the untyped calculus with the invention of domain theory.
A: In the standard set theoretical setup, a function cannot have itself as an input. This is because the rank of the function is strictly larger than that of its inputs and outputs.
https://en.m.wikipedia.org/wiki/Von_Neumann_universe
So when they say id(id)=id, it is meant in a more algebraic sense where composition is really just a kind of multiplication or binary operation. 
A: First, these are not functions. These are lambda expressions that can be artistically interpreted as functions. If you want to model lambda theory in a theory where functions are a part of the discourse, it's a different story. As it was told above, one solution is to model lambda in Set Theory. Sets have functions defined, so lambda expressions would be represented as functions. The problem is, of course, that it's impossible to model without tricks. The solution was provided by Dana Scott in the '60s. See, e.g., http://www.users.waitrose.com/~hindley/SomePapers_PDFs/2006CarHin,HistlamRp.pdf or watch him at LambdaConf, https://www.youtube.com/watch?v=mBjhDyHFqJY&t=2s
