The RTC won't have enough resolution for what you want to do -- it's a 48-bit counter with a resolution of 1/1024 second.
I'd recommend looking at the documentation for the BL4S100 or BL4S200 series SBCs. They're newer products, using an updated compiler, and include an advanced I/O processing chip called "RIO". I don't know all of the capabilities of that chip, but the documentation should allow you to determine if it meets your requirements.
I know that it has pulse counting capabilities for sure, and I think it can measure pulse width as well. You might want to download and install Dynamic C 10.72A, then look at the files in Samples/BL4S1xx/DIO to learn about pulse capture capabilities.
If this is some serial protocol, like IrDA or HDLC, the Rabbit's UARTs might be able to decode it for you.