Quick answer to the compatibility question: They're not really compatible. It doesn't blow up, but you'll get some anomelies. However, the latest
v.28 semi-official beta of R.E.D. has quite a few more units than E.U. so it shouldn't matter. There's a few units that Ethnic Units has that R.E.D. doesn't, but that's by choice.
Ethnic Diversity has
even more units, but like Ethnic Units is only vanilla scale.
Long answer to the historical question: R.E.D. stands for Regiment and Ethnic Diversity, which refers back to the previous Civ4 incarnation of this mod ("Regiments, Formations, Air Force, and Ethnically Diverse Units") which started out by combining several Civ4 mods, including Ethnically Diverse Units, and Regiments.
These mods all use basically the same set of available units. The mod creators don't make the models; they just compile the stuff that other talented people have shared.
Ethnic Units was based on an old version of this mod (v.11 if memory serves), before Gedemon switched to the SQL table format it uses now. Ethnic Units backed out the formation and scaling changes that R.E.D. always included. For a while it included more units, partly because Gedemon had stopped updating for a time, but also because he preferred not to (arguably) shoehorn in a unit model that was available but didn't quite fit. Gedemon eventually did make some updates that closed the gap considerably. I took over for a little while and my
v.28 semi-official update eclipsed Ethnic Units by a wide margin since BrutalSamurai22 hasn't updated E.U. in years.
Ethnic Diversity is another fork that I introduced during one of Gedemon's absences, prior to Ethnic Units, but I discontinued it when Gedemon came back. I reintroduced it again recently (what, 6 months ago?) adding even more units than v28. It's uses vanilla scale, but R.E.D. scale was planned to be added as a separate addon. However, that hasn't been completed yet and my time is extremely limited at the moment.