is it generally possibly to write and run my own firmware on the bitforce single?
Well, I would not say that it's generally not possible.
BFL has not publicly documented the design and has at this point in time no interest that clients running not BFL released logic on the singles.
However it's a FPGA based product. How to use the JTAG connectors in the single is already known. If you want to use the microcontroller or usb also you have to do some reverse engineering.
Other products are open source or have at least a good documentation (schematic etc.) to use it for other tasks also.
I don't understand why BFL keeps so many information as a secrete and doesn't use the power of the community to improve the products. BFL's main advantage is the good source for the FPGA's.