Skype on iPhone - does phone have to be awake / unlocked to receive calls? I have just installed the Skype app on my iPhone 5 running iOS 10.3.3. 

To recieve Skype calls does the phone have to be awake/unlocked?

Does the Skype app have to be open/launched? I periodically double click the home button and close / swipe up all open apps. I presume if I do this to Skype I can't receive calls?

To receive calls does the Skype app have to be the currently open app? Or can I use other apps, with Skype open in the "background"?
Skype doesn’t have to be open. I have tested it. There have been known issues with this which you may be experiencing.

In iOS 11 a feature was introduced that enhances third-party calling, so updating should fix the issue.