Can an app make phone calls in the background
I am an Android Application developer. I had a requirement to make phone verification. In that case we had to make a telephony call to our server. But we could't achieve that as Android system does not allow us to make a phone call without users interaction. The best we can do is populate the phone number in the default dialer app. In any case user has to press call button. So I would say No app can make a phone call without your permission. 

But

We can send SMS in from background and it will cost you money. For this you will see something like this before installing app: