Hmm well let's see... As far as dominance in my collection goes, Classic Rock is the strongest. But a lot of that is material I bought before I listened to anything post-1980; I was desperate, so probably 25% of it is stuff I never listened to very much and may well never listen to again.
Classic Rock is definetly my favorite outside of metal what with artists like Pink Floyd, Neil Young, The Doors, and Tom Petty. But I've been listening to most of that stuff for many years and so I'm not crazy about it, I just like it. Grunge is the non-metal genre that is "coolest" to me but when I try to sit down and listen to it again it often seems archaic (iow too soft), with the exception being With The Lights Out.
I listen to a lot of folk, but mainly coming from Classic Rock artists such as Bob Dylan and Neil Young. Oh, yeah, and how could I forget rap? Eminem and Immortal Technique are 2 of my current favorite artists out of any genre. Rap is very hit or miss for me, even my 3rd and 4th favorites (Dr. Dre and Necro) are artists I love but only occasionally listen to. I think rap might have the greatest lyrics of any genre, but I find that it loses its impact very quickly.
My basic rule is... anything with a guitar and/or lyrics which doesn't fall under the category of "easy-listening" is cool with me. Not literally, but I think eventually I'll probably tolerate literally anything with those criteria.
As for the connection between my favorite non-metal genres and my favorite metal genres... I guess there's kind of a connection, to an extent. I'd call Pantera and Metallica two of the most Classic Rock-esque metal bands except that I'm not a diehard fan of Sabbath, Priest, Maiden, or Motorhead. Not *yet*, at least. Some may disagree but I'd say some grunge bands connect to 2nd wave black metal with the mutual minimalism and rawness. I'd also say that Pantera's lyrics are a lot like Eminem's.