If I have a device that transmits data over Bluetooth, is it possible to get at that data via a different device with my own custom code?
(I would rather leave this as a comment, but don't have enough rep)

If your computer has bluetooth transceiver you can use a packet sniffer program like Wireshark (Unix and Windows) to investigate the packets and try and understand the data transmission protocol.

http://wiki.wireshark.org/CaptureSetup/Bluetooth

Be aware though that you will be looking at packets meant for the data transmission layer (See the OSI Model) which means that there will be data bits in the packet relating to the transmission hardware along with the data, which will be encoded in a specific way for use with software.