While it's never explicitly said one way or another. I've seen instances of Khajiit speaking in first person in games. Admitably, many of these are generic dialog that all races use, so in these cases, it could be that Bethesda just didn't include specific dialog for Khajiit characters in these cases. But I seem to recall even some lines written specifically for Khajiit characters being delivered in first person. This seems to be much more common in Oblivion than Morrowind, but I think I've heard some generic Khajiit greetings in Morrowind written in first person too.
My theory on the matter is that speaking in third person is a speech habit that's connected to the Khajiit language. And many Khajiit, despite knowing Cyrodiilic, or whatever the language everyone is speaking in the game, still retain some habits from their own speech. I certainly don't see any reason why they COULDN'T speak in first person. But many Khajiit are not used to doing so.
In general, though, when role-playing Khajiit characters, if I think using in-character terms, I tend to try to form my thoughts in third person, which can actually a bit of a challenge as I'm generally used to thinking of what I do in the game as things
I do rather than what my character does. Of course the journal is still written in first person, but I don't usually worry about that too much, as I generally only think of it as a means to help me remember what I'm supposed to do for my quests, not as an actual record of my character's thoughts on what happens in the game. I think that's what Bethesda meant it to be anyway since most of the journal entries seem to be written in a way that doesn't actually bring the character's personal oppinion into the matter, leaving it up to players to decide what the character thinks.
I'd say dunmeri, as the dunmer are no longer chimer. But I'd also assume a large number of NPCs in MW are speaking Cyrodiilic, especially among those who are not dunmer, the Hlaalu, and the imperialized dunmer.
I don't know. I'd think that if any of the characters in the game were speaking a different language from what others use, the game would tell us somehow, like by using a different font or something, anything that would indicate that they're not speaking in the same language as anything else.