I think your rank is your rank. Stuff is different in Gi and in no Gi, yes, but you have been ranked nonetheless. Going to roll in the Gi? Learn the grips and lapel chokes, get a feel for it. Rolling no Gi? Learn to clench and be tight in transitions rather than depending on grip as much. We have a blue belt that comes in. He is from Brasil and is a world champion too, I believe. He has trained mainly Gi, we are a no Gi school. He is still a blue belt. You can feel it.
If you want to wear a white belt out of respect, by all means. I would prob just pack both if I was going to train Gi somewhere and ask the instructor. All instructors are different. Some are lax while others might be majorly offended that you wore your no Gi rank.
If you are a black under anyone and you want to be belted in 10p, you start as a brown belt. As far as the rashguard situation, is there an official rule on that at HQ? I would assume a black belt visitor wouldn't wear a 10p black belt rank rashguard, but I wouldn't see a problem with a ranked rashguard by Gracie, Jaco, ..or someone else.