How big or small is a reference frame in Relativity? What exactly is a frame of reference? Does it have an objective existence and if so what is it? 
What's the size of a reference frame? Is a reference frame the same size for a stationary frame of reference and one that's accelerating?
What's the smallest frame of reference? Does a photon or electron have a reference frame or does this just apply to classical objects?
 A: A frame of reference is just a momentary choice for the direction of time (and therefore, for what constitutes space, as space is orthogonal to time).  A choice of reference frame is not conceptually different from choosing a direction for "up" on a map.  Once you have chosen "up" at some point, the notions of left and right become clear.
All of us are going through spacetime, and the "direction" we travel through spacetime defines how we split things up into space and time.  A reference frame is merely a choice of time direction against which to make measurements.  It no more "exists" than it would make sense to say a compass rose exists on a map.  You can write it down or describe it, but it is not a thing in and of itself.
A reference frame has no size.  Strictly speaking, a choice of frame at a point says nothing about what frames might be used at other nearby points.
Any timelike object can be said to have an instantaneous reference frame.  Photons are different in that their trajectories do not split up spacetime into space and time.

What is spacetime?  Again, I return to the analogy of a map.  On a map, you have many different locations and features.  Each point on a map represents a location.  Spacetime is a map for the whole history of the universe, in which a point in spacetime represents not just a location, but a particular location at a particular time.
On a map, you can freely choose which direction is "up" (and therefore what is left or right).  There are many choices of map coordinates that you might make, depending on what is convenient, but these choices do not change or alter the arrangements of locations in any substantive way:  all the locations are the same, you're just describing them with numbers differently.
For spacetime, a choice of reference frame is a choice for the direction of time and the directions of space as a result.  This means that points in spacetime can be described with different combinations of space and time coordinates, each set of coordinates unique to a particular reference frame--to a particular choice of directions--but these choices do not in any way move events around in spacetime.  The coordinates are different merely because one chooses the directions used to measure those coordinates differently.
So, what is spacetime?  A model for how events in the universe fit together in a way that can be described with coordinates for space and time.
Does it have an objective existence?  I answered the previous question trying to emphasize that reference frames are all math, in the sense that you cannot reach out and touch one.  You can't reach out and touch spacetime either, but the model has been successful--especially in how objects influence spacetime and therefore influence the paths of other objects.
