Just had this situation when Zylah (Fly ability) was on the board for the first time.

Is a Flying hero allowed to open chests right under the bottom of a blocking monster, just like an Acrobat can do?

I don't see any way that Fly could possibly be different from Acrobat for such purposes. It's not like the Acrobat card explicitly says "and you can open chests when monsters are standing on them," that's just a side-effect of the fact that you can move onto chests when monsters are standing on them.