Pros: You can develop the architecture exactly how you want
Cons: You waste loads of time, when an existing core could have done everything for you
What do you think the distinction between a microprocessor and a microcontroller is? Traditionally, a processor was a chip that had few peripherals or memory built in and was pretty much useless on its own, while microcontrollers bundled RAM/ROM, timers, ADCs, GPIOs, special I/O peripherals and so on. In an FPGA it's reasonably straightforward to build a 'processor' as the core of your design, and then attach only the peripherals you need for a particular application.
Xilinx's EDK is a good example of this - the MicroBlaze processor has no I/O capabilities on its own, but you can attach GPIOs, memory controllers, Ethernet MACs, and so on using a standardised bus.
In general, you can reimplement any processor core on an FPGA. The challenges are that detailed design documents might not be publicly available, or the specifications can be incomplete or even contradictory. For 100% compatibility, you would need to write test programs that exercise ALL functions of the CPU in EVERY conceivable combination, and ensure that your own version replicates the physical device completely. As you might imagine, even with a moderately complicated CPU, this is nearly an impossible task to do unless you have (literally) thousands of years for all tests to run.
If the 80251 is broadly compatible with an 8051, your best bet would be to modify one of the open source or public domain 8051 cores.