When How to Start Submitting Apps to The iOS App Store Written Using The Swift Programming Language

When can we start submitting apps to the iOS App Store written using the Swift programming language?

In the fall when IOS 8 is officially released. September 2014.

From the apple website:
"And when iOS 8 and OS X Yosemite are released this fall, you can submit apps that use Swift to the App Store and Mac App Store."

reference: https://developer.apple.com/swift/

Has anyone been able to submit Swift apps to the App Store?

No! No one has submitted their Swift app in the Store, because it's not officially released yet, as you yourself mentioned.

You can begin using Swift code immediately to implement new features in your app, or enhance existing ones. New Swift code co-exists along side your existing Objective-C files in the same project, making it easy to adopt. And when iOS 8 and OS X Yosemite are released this fall, you can submit your apps to the App Store and Mac App Store.

From http://developer.apple.com/swift

How to run my app locally without submitting to the app store

You do not need a developer license after Xcode 7. You can use Free Provisioning:

Launch Your App on Devices Using Free Provisioning (iOS, watchOS)
If you don’t join the Apple Developer Program, you can still build and run your app on your devices using free provisioning. However, the capabilities available to your app, described in Adding Capabilities, are restricted when you don’t belong to the Apple Developer Program.

In Xcode, add your Apple ID to Accounts preferences, described in Adding Your Apple ID Account in Xcode.
In the project navigator, select the project and your target to display the project editor.
Click General and choose your name from the Team pop-up menu.
Connect the device to your Mac and choose your device from the Scheme toolbar menu.
Below the Team pop-up menu, click Fix Issue.
Xcode creates a free provisioning profile for you and the warning text under the Team pop-up menu disappears.
Click the Run button.
Xcode installs the app on the device before launching the app.

In your case, you can perform this process for your iPad by connecting it to your Mac. Then repeat the process by unplugging it then plugging in your wife's iPad.

More details can be found in BoltClock's answer I copied this from in the related question

How to know the App ID from the App Store before distributing the version 1.0 of an App with Xcode?

If I understand your question correctly, you can just create a new entry in App Store Connect and you will get the app id in the "App Information" menu item (under "General Information"/"Apple ID").

You don't need to have the first version of your app distributed, this id won't change.

Do Swift-based applications work on OS X 10.9/iOS 7 and lower?

I just tested it for you, Swift applications compile into standard binaries and can be run on OS X 10.9 and iOS 7.

Simple Swift application used for testing:

func application(application: UIApplication, didFinishLaunchingWithOptions launchOptions: NSDictionary?) -> Bool {
    self.window = UIWindow(frame: UIScreen.mainScreen().bounds)

    var controller = UIViewController()
    var view = UIView(frame: CGRectMake(0, 0, 320, 568))
    view.backgroundColor = UIColor.redColor()
    controller.view = view

    var label = UILabel(frame: CGRectMake(0, 0, 200, 21))
    label.center = CGPointMake(160, 284)
    label.textAlignment = NSTextAlignment.Center
    label.text = "I'am a test label"
    controller.view.addSubview(label)

    self.window!.rootViewController = controller
    self.window!.makeKeyAndVisible()
    return true
}