Can an MFRC500 read a 125khz RFID card?
In answer to 1) no - these readers do not contain broad spectrum tuneable oscillators and receivers - they are optimised to the job they are designed for.

In answer to 2) yes, but why bother? The RFID protocol is tricky, requiring at very least a programmable microcontroller to decode / encode and an RF amplifier / receiver to communicate. Solution-specific ICs at affordable prices are excellent news - why would you want to implement something else?