I doubt that the intended recipient of the note matters, otherwise Moody wouldn’t have set fire to it as soon as Harry had read it. I expect if someone else had read the note, it would be the same as them having overheard Dumbledore telling Harry the location. If the intended recipient were of importance, I’d expect that the note would have begun, “Dear Harry”.
Not if the notes were written for a particular person. If Dumbledore writes a note for Harry, and Moody delivers it, then it is Dumbledore that has divulged the secret. If Moody reads the letter, assuming he was already in on the secret, then it remains a secret. If Moody reads the letter to Tonks, assuming she was already in on the secret, then it remains a secret. Yet because he was not the master of the secret, Moody would have been unable to pass the note on to say...Lavender Brown. Either his arm would not move, or the parchment would not unfurl, or whatever, but it wouldn't be possible.