As far as I know, essentially WIFI direct is enabled by a soft AP. My question: can the underlying WIFI module work as a WIFI station and as a WIFI direct endpoint simultaneously? That means, we hope that the WIFI module can communicate with a mobile phone using WIFI direct, and at the same time connect to a router as a WIFI station. Is it possible? and how widely and strongly is this case supported in technology?
You can if your wifi driver supports multi-role. This feature, available in some wifi chip (e.g TI's wl1271), allows you to use one wifi chip for different roles.
We use this feature in one of our product to maintain an access point while still being able to connect to another wifi station at the same time. This product runs on Android 4.0.4 with a OMAP 3.0 Linux kernel.
For more informations on this you can check if such a driver exists for your chip and take a look at this post I created on Stackoverflow where I give the details of our implementation using our wireless chip.