I can vouch for the HD-600/RA-1 pairing, FWIW. I also second the recommendation to use batteries. I've got an older Grado amp (HPA-1) that can switch between AC and battery power on the fly. There is a noticeable drop in the noise floor using battery power. I've heard the HD-600 with EMP, but not enough to recommend one way or another. A friend and I, doing comparisons, tended to prefer a Moretto HAP-03 with high impedance headphones, and the EMP with low impedance cans.
Of interest is that in doing the comparisons, I was able to put the same output tube on both amps, and very similar gain tubes (same brand, 12AU7 vs. 12AT7). The EMP's signature, as will any tube amp, is going to vary with the tubes chosen.
The RA-1 tends toward the warm side of neutral. It tends to be slightly euphonic, rather than analytic. So, be aware that it will impart its coloration to the sound, as will just about any amp or headphone.