(1) The type of hops you would use if you could only use one type for all of your brews.
Another vote for Cascade
(2) Why did you choose this hop?
Very versatile, at least within the context of "American" versisons of styles. High enough AA% to use for bittering, and a nice, clean, classic flavor and aroma for IPAs, APAs, etc. If I had to limit myself to one beer "grouping," it would be American beers; there's enough diversification among APAs, IPAs, Texas Browns, Americanized Stouts, etc. to keep things interesting, and Cascade can work in all of those.
(3) What limitations would this one hop type put on some of your brews?
Obviously, traditional English or German ales/lagers would be very difficult, although I would probably give some malt-driven German styles a try and hope that ALL of the flavor/aroma was out with a 90 minute boil. Wouldn't be ideal, but might work OK. Tough to make classic English ales, but I haven't been terribly successful at brewing these regardless, and this would just force me to develop my own American interpretations of something like a mild ale (which I think should be done, regardless).
Belgian beers can, in some contexts, be viewed as a little more forgiving of ingredients (not as strict with the "rules"), and since many of the Belgian styles I like are hop-light and require a 90 minute boil (Pilsner malt), I could probably get by with Cascade in there as well.
Really, the beers out there that are most hop-sensitive are American and English styles (IMHO), and I'd take American beers over English if I could only pick one. German and Belgian styles tend to not emphasize hop flavor and aroma, so I suspect I could cheat with a non-traditional hop more easily than if I were to try and brew an American IPA with something like Tettnag.
(4) What types of hops do you currently use for your most-brewed recipes? (or if you don't really repeat recipes, which type of hops have you been using again and again?)
I've got Cascade, Tettnag, and Fuggles in bulk, so I've been using those wherever I can in most recipes. I've got a few other things like Amarillo and Centenial that I've used a fair amount of, but those first three hops have been my default choices for American, German, and English-style beers (if I had my druthers, I'd have Hallertau instread of Tettnag for the Germen and EKG for the Fuggles, but I bought what was available at the time).