Swift has great support for interfacing with C and Objective-C. The Swift compiler ingests any C headers you provide and will generate the right machine code to call those C functions, no need for you to write any manual bindings like in other languages. It is not foolproof- it can only decipher constants from the C preprocessor, for example- but works very well.
Right now, there is even work under way to go the other way more easily, ie write your code in Swift and call it from C. 
Oh, there is definitely a Swift runtime, you can find its source here. It is written in a mix of Swift and C++. I was merely pointing out that it builds on top of your OS C library on most platforms, except maybe less for the embedded targets.
Not that I know of, but difficult to keep track of all the Swift articles and blogs out there. 
Use your favorite search engine and see what you find.