This is mostly a +1 to everything since Ashitaka functions very similarly to Jon in this scenario. When it comes to some things more unique to him, Ashitaka doesn't do too much expounding on wherever he is from, instead leaning on the novelty of his not-being-European to avoid too many intricate details. As for what they will call him... he knows just as much about French naming conventions as they know about Japanese ones, so he will just accept whatever name they give him. Ashitaka feels that being as genuine as possible and seeking to gain information from Lafayette (and other guards, if possible) by proving dependability is the best route, though if it ends up running dry, he will agree to some sneaking and subterfuge (though a little begrudgingly). He would work together with Jon to increase their chances of success, having one or the other cause distractions if necessary.

Ashitaka will handle his duty as guard with seriousness (to a point of severity...), but (if this comes up at all) he will speak to anyone with his usual openness and understanding, though tempered by degrees of respect for those hierarchically above himself. I can specify any reactions or responses if necessary.

The only other thing I'd tack on is that he would just claim any conversations with Jon or Dany to be for the safety of the estate and the royalty. He's... not a very good liar.