Is it possible to develop an iOS framework with storyboard?

One way of creating UI Framework is to create VCs with nib file. If you have segues defined within the framework, then go for storyboard.

If you are using storyboard, expose UI helper class wherein you can instantiate VCs from storyboard. For eg:

public class Utility {

    public static func getHomeVC() -> UIViewController {

            let storyboard = UIStoryboard.init(name: "HomeScreen", bundle: Bundle(for: self))
            let homeVC = storyboard.instantiateViewController(withIdentifier: "FrameworkHomeVC") 
            return homeVC
    }

}

Yes.

The storyboard is there just to hold your UI. You can use it to instantiate ViewControllers.

The code is separated into code files.

[iOS Framework] is just a container for sources and resources(like .storyboard).

When you build framework with .storyboard you get something like

Here you can find .storyboardc which is compiled .storyboard

[Get storyboard from a framework]