The fields and waves associated with a transformer
An electric field is associated with a fluctuating magnetic field. Is
  the electric field at right angles to the magnetic flux lines?


The simple answer is yes as per this diagram from here: -



For a transformer, the changing flux is contained within the core and the windings that surround the core limbs "collect" the field that is at right angles to the magnetic flux lines. Two turns of a winding collect twice the voltage etc..


  Is this magnetic field actually an electromagnetic wave? What part, if
  any, do electromagnetic waves/fields have in the operation of a
  transformer?


No. It is a magnetic field (and an associated electric field) and no part of understanding the operation of a transformer is relied upon by using electromagnetic wave theory. Electric and magnetic fields are not forming an EM wave because they are temporally shifted; it is the rate of change of magnetic flux that delivers voltage; BUT in an EM wave, electric and magnetic fields rise and fall with time syncronicity: -



This doesn't happen with a transformer because the induced voltage is \$N\dfrac{d\Phi}{dt}\$.


  Do electromagnetic waves travel down the wire?


If you dig deep enough you will find that all currents and voltages can be thought of as EM waves and, as such, all wires can be thought of as transmission lines. Do you need this to understand "regular" transformers? 

No!

Do you need this to understand RF transformers? Sometimes when in the upper VHF region and beyond and certainly as you approach the microwave range it is important to understand what effects can happen.