Lee Odden interviewed Adam Lasnik of Google the other day. Adam Lasnik is one of the folks at Google responsible for being the voice of the webmaster. His day job is to help webmasters with ranking and indexing issues either through communication or other means. I find the read interesting and helpful - it really shows that Google cares.
As for what questions annoy me the most? There aren't any specific ones that I find particularly frustrating. Rather, I do occasionally grow weary with two types of questions:
1) Questions that are clearly answered in our much-improved Webmaster Central, via a quick search of our Webmaster Help group, or questions that would also be likely answered via use of our Webmaster Tools. There's no such thing as a stupid question, IMHO, but lazy questions… well, that's a different story.
2) Accusatory “questions.” I suppose I need to get some thicker skin, but it stings when people imply that we either don't care or — worse — that a relationship between Webmasters and Google must inherently be adversarial. Every time I've spoken with Larry Page and Marissa Mayer they've made it unequivocally clear that being mindful of Webmaster concerns is something resonating not just in Search Quality, but from the very top of Google.