Let's see if i can summarize.
The mainstay of SR is that c (the speed of light in vacuum) is the same in all inertial reference frames. This has some interesting implications--basically, nothing else is the same. Distances, time, "simultaneous" differ from one reference frame to another. There are a handful of simple equations that describe how to translate from one reference frame to another; regardless, translations from one reference frame to another are non-intuitive, and are excellent opportunities for introducing mistakes in calculations.
The equations for relativistic doppler shift are different from the equations for classical doppler shift.
SR has no c-regulators; it is in direct opposition to older, "classical" theories that would require a c regulator.
Time dilation is real and has been demonstrated.
It seems that the validity of the relativistic doppler shift formula could be demonstrated; off hand, i don't know of any experiments that have demonstrated this, but i haven't looked for any.