The topic of diamond wheel/HHS comes up pretty often and is widely stated diamond is for use on carbide, with some more knowledgeable? poster putting forth the 'reasons'! I personally dont know, but taken as a whole, I do abide by the majority in this case and use my wheels for only carbide. Obviously a diamond will cut HHS, but apparently is best utilized on carbide.
As for cooling. I dont recall ever reading of a home shop using cooling on a diamond wheel. Thinking about several of the common home shop tools using a diamond wheel - TCG, bench grinder, carbide grinder (HF), they arent set up for any real coolant use - maybe a drip bottle - and even that seems more designed for the use with HSS.