Cor and Rabadash are contemporaneous royalty of neighboring kingdoms -- it would be odd if they never met officially... and since Rabadash can't leave Tashbaan, obviously Aravis and Cor must go to Calormen. (I am pretty sure they only go once after Lune's death, though.)

The idea of Corin pressed into service as his father's or brother's ambassador is a thing of great and terrible beauty. It works fine in Narnia. It works passably in the neighboring mountain lands of Mergandy, Sarovence, and Telmar, if only because they are reluctantly used to him. It may even work, occasionally, in the various island kingdoms. But in Calormen? No. That is a disaster waiting to happen. :-)