MM,
1. If one holds that it is prohibited to walk on an part of HHB because of tumat met then walking there is not behaving like a Jew. In any case, IMHO before we can ask to go up on HHB we have to learn how to act in shul, the mikdash me’at.
2. I would shake hands with (male) goy who says “Am Yisrael Chai” even not on HHB.