MMX didn't add new registers. It uses the same FP registers as those present all the way back to the Intel 8087 in 1980. You could say the MMX registers are older than the 32-bit x86 architecture, and therefore older than EAX, EBX, ESP etc.
Not exactly, especially in the context of this issue. You could use the MMX registers as 'traditional' registers (or at least a scratchpad), not quite like the x87 stack.
There are 8 64-bit MMX registers. To avoid having to add new registers, they were made to overlap with the FPU stack register. This means that the MMX instructions and the FPU instructions cannot be used simultaneously. MMX registers are addressed directly, and do not need to be accessed by pushing and popping in the same way as the FPU registers. [1]
