Jimmycricket, I had the same question. My view is that not only would all the stars burn out, but all the black holes would evaporate by Hawking Radiation before you crossed the horizon, including the one you are falling into.Would the universe end if you entered a black hole? What I mean by this is that due to time dilation would time elapse so fast for the universe outside the black hole relative to you inside it that all the stars would burn out and all that wouild be left would be other black holes?
As for separating the time dilation effect into kinematic and gravitational, that's easy. The gravitational effect blue shifts the outside universe without bound as you approach the event horizon, but your speed approaches the speed of light as you approach the event horizon thus red-shifting the universe without bound as you approach the event horizon (all stationary people (dr = 0) that you pass on your way to the horizon will be waving at you very slowly because of their speed relative to you, which approaches that of light as you near the horizon). Taking the limit of these two "infinities" as R approaches 2m yields 1+√2m/R as stated by George Jones.