Xcode 5 is behaving just fine, and I\'m able to build and run apps on my iOS 7 device and in the iOS 7 simulator.
However, I do want to support iOS 6.1, but I don\'t
Xcode5 neither has 6.0 SDK or 6.0 simulator. But what you can do is u can copy both 6.0SDK and 6.0 simulator from your previous Xcodes (xcode4.5 for 6.0 and XCode4.6 for ios6.1) You can find the simulator at /applications/xcode.app/contents/developer/platforms/iphonesimulator.platform/developer/applications/iphone simulator.app
And SDK at /applications/xcode.app/contents/developer/platforms/iphoneos.platform/devicesupport/6.0
Copy the entire folders for simulator and SDK in the exact same path of XCode5. Note: to go the the Contents of Xcode.app just do a right click on the .app file and you will get an option"Show Contents".
Once u are done copy pasting, restart ur XCode5 and u can see 6.0/6.1 simulator as an option :)