Is it possible to make a computer behave as a bluetooth HID device?
It is definitely possible on Linux. Some time ago I found this project:
http://nohands.sourceforge.net/index.html
They emulate a full-blown headset with audio and keyboard controls on the Linux bluetooth stack. If they can emulate something like that, you would probably be able to emulate something simpler like a keyboard.
It is possible, however I don't think I'll be able explain it very well and I don't know the entire answer. A BT HID device works as a server and waits for connections to come to it. In linux, using the bluez stack, first you would have to advertise the HID service for other devices to see. I think you do that using the sdp.h and sdp-lib.h header files(the second header maybe called something else, I'm on a windows computer and can't check). So you would have to add the HID service record to you computer for other devices to see it. You would have to create a program that first adds this service to the record, then waits for other devices to connect, then handle the pairing process, Bluez might handle this for you, or you might have to do some things to it, I'm not quite sure. You should also read the Bluetooth HID Spec found at the http://www.bluetooth.com/English/Technology/Building/Pages/Specification.aspx site. This document contains the details of the SDP record relevant to HID. Also the book Bluetooth essentials for programmers is pretty good to introduce you into bluetooth programming
I would like to have given a more concise answer, with more detail, but that's all I know ATM. I am also trying do something similar, but spare time is so hard to find ;) I'm also not on my Linux box and can't check all the details. If your are still interested, let me know and i'll try to expand my answer.