Khajiit are commonly used. Bosmer might be, too. The style of the assassin may vary, though. I don't know if the Thalmor have ever had any assassins of the Altmer race, though. Possibly before they forced Valenwood and charmed Elsweyr into joining them. Hit squads don't really count because those are just for gameplay purposes or just are only used when they are threatened by someone dangerous. The Dovahkiin is dangerous to them, so yeah. If I'm wrong, someone correct me.
As for how the killings are done, magic may be used. May not. The Khajiit are not usually gifted with much skill in magic. The Bosmer have more, but they are elves. Of course they are more likely to. I don't think the Thalmor would send out any of the less seen breeds of Khajiit for any assassinations.
The main reason you won't see a Dagi-Raht or Dagi, even though they are gifted with magic unlike most breeds of Khajiit, is because that is a breed most races aren't used to. I could see them using Ohmes and Ohmes-raht Khajiit assassins more often than Suthay-Raht ones. The main reasoning for this is due to ohmes being hard to tell apart from a Bosmer. Ohmes-Raht have a tail, but you could still somewhat mistake them for Bosmer.
Bethesda might have changed the lore on Ohmes being more commonly seen due to the breed in Morrowind, though. The Suthay-Raht may be more commonly seen in lore now. It just seems impractical to have the Suthay-Raht being used as assassins in places like Skyrim. You will stand out way too easily. An Ohmes could be confused with a Bosmer and blend in better.