How was matter in other parts of the universe affected by the Big Bang and inflation in an infinitely sized universe? I'm trying to understand how the idea of an infinite universe with presumably infinite matter works with the Big Bang and inflation.
I understand that if the universe is infinite, then it's always been infinite, and that when we talk about the initial universe being very very small, dense and hot we are referring to the observable universe (correct me if I'm wrong).
So let's say in our infinitely sized universe, there's a galaxy (Graham's Number)^(Graham's Number)^(Graham's Number) light years away, or some huge distance like that.
Would this galaxy have also been very close, hot and dense with our observable universe? If so, what if I just keep increasing the distance since I've assumed the universe is infinitely big? 
And if not, would it have had its own separate big bang?
So my question is how really really far away regions of the universe were affected by the big bang, if the universe is infinite since they couldn't have all been close to each other in a very hot and dense state.
 A: 
Would this galaxy have also been very close, hot and dense with our
  observable universe?

(This is more of an extended comment than an answer)
Note that when 'running the movie projector in reverse', one would eventually get to the time before this galaxy existed but well after the universe became transparent. Eventually, one gets to the time before galaxies could exist.
Also, it isn't clear what you mean (or are thinking about) by "very close". As safesphere's answer (in a comment) points out, it's better to think about this in terms of whether or not that galaxy (or the 'stuff' that eventually makes up that galaxy) is always outside of cosmological horizon regardless of how 'close' that might be.
As an aside, but possibly helpful in thinking about this, there is a cosmological model, Roger Penrose's Conformal Cyclic Cosmology (CCC), where the 'Big Bang boundary' of our universe can be identified with the 'infinitely expanded' state (where all mass has decayed such that there are only massless fields) of the previous "aeon".

Penrose's basic construction is to connect a countable sequence of
  open Friedmann–Lemaître–Robertson–Walker metric (FLRW) spacetimes,
  each representing a Big Bang followed by an infinite future expansion.
  Penrose noticed that the past conformal boundary of one copy of FLRW
  spacetime can be "attached" to the future conformal boundary of
  another, after an appropriate conformal rescaling. In particular, each
  individual FLRW metric $g_{a b}$ is multiplied by the square of a
  conformal factor $\Omega$ that approaches zero at timelike infinity,
  effectively "squashing down" the future conformal boundary to a
  conformally regular hypersurface (which is spacelike if there is a
  positive cosmological constant, as is currently believed). The result
  is a new solution to Einstein's equations, which Penrose takes to
  represent the entire universe, and which is composed of a sequence of
  sectors that Penrose calls "aeons".

