Is it possible to create responsive, mobile-style apps without coding directly for iOS or Android with the same html/js/css stack that is used for major browsers?
Is it possible to create responsive, mobile-style apps without coding directly for iOS or
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Have a look at these books, they really helped me get started with developing for mobile using web technologies:
Building Android Apps with HTML, CSS, and JavaScript
Building iPhone Apps with HTML, CSS, and JavaScript
As others have said you can make a web app styled for mobile. There are some nice framworks to help you with styling and things like nice animated swipes etc. I would recommend looking at jQuery Mobile and Sencha Touch.
For sencha there is a list of getting started resources in my answer to another stackoverflow question here.
If you you want to make your web app native (and have access to features like the camera and accelerometer) I would recommend PhoneGap, but there are other frameworks already mentioned like Titanium.
Phonegap can be used along with Sencha Touch or jQueryMobile and has lots of resources to help get you started e.g. the API docs and getting started guide (check out the phonegap website for others)
Hope that helps!!