Can an Android device pretend and connect as a Bluetooth controller? Can Android devices connect as a device to, say, a PC and act as a gamepad or similar device? Is it allowed and doable in code without root access?
I tried this code: Connect Bluetooth devices. But it can not get it to pair.
Short answer, yes.
I recommend looking at Google's example project on Bluetooth communication BlueToothChat initially. Based on what you've written, I think you're missing several important steps.
You need all of the following components / steps:
If you have access to the device your attempting work with as a controller, then you can create your own communication setup. Define a set of parameters to write and read on both sides that emulate the functionality you're looking for.
If you do not have access to creating your own read/write setup, or you want to use standard interaction methods for the industry, then the best bet is the Bluetooth HID Device profile mentioned by @Morrison Chang.
The HID profile basically establish an industry standard of required device features, read/write formats, combinations of features equaling a controller type (ie: "this is a keyboard"), and data mappings for what read/write numbers mean on either side of the client / server connection.
From the linked comment thread, the Kontroller project on Github has source available for your peruse.
There's also a reasonable Intro to Bluetooth HID that covers some of the topics on what Bluetooth HID is actually doing.
Finally, also recommend looking at Google's own BluetoothHidDevice page.