Don't know about your first issue, can't really say that I've needed to delete those and come to think of it, I don't know how Maya can define a surface thats missing border edges :x Maybe someone else has a solution.

Your second issue, turn on backface culling, select the object, Display>Polygons>Backface Culling, there are also option for this right beneath that. If you use it enough you might want to assign it to a hotkey or something. If you turn it on in the prefs it'll affect all your poly geometry upon creation (which personally I find annoying as hell).