Hello, I'm interested in this project . If I understand, you need an application that once was started to be showed in full screen and to not allow access to other applications(just allow to call those 4 numbers)
I already have some experience In processing of Incoming and outgoing calls and custom call screens.
I read all the requirements and know how to do it (Show your screen above other app, initiate, answer, end call), I will be glad to collaborate further.
I'm working in Java , Eclipse , use android SDK.
Contact me to discduss the details.