Is there a necessity for the mouse to know a different protocol or a
different pinout to be able to be connected to a PS/2 port?
Yes, it is necessary for the USB mouse to support the PS/2 protocol to be connected to P/2. I can imagine there's someone somewhere that made an adapter somewhere that proves this false but that's a rare exception that would cost more than just buying another mouse and therefore be a personal design challenge than any viable mass produced product.
Many vendors provide a USB-to-PS/2 adapter, but some don't.
From what I know, PS2 and USB interfaces even have a different number
of pins. How would such an adapter work?
Of the six pins on a PS/2 port only 4 are used. For reference see this: https://en.wikipedia.org/wiki/PS/2_port
An exception (pointed out in the linked article) would be some laptop PS/2 ports that use the normally unconnected pins to allow for a splitter that can split the single PS/2 port into two PS/2 ports.
An educated guess is that the mouse knows which kind of port it is connected to by switching polarity on the power pins, put the +5V on one pin and the data pins "talk" USB, put +5V on the other pin and the data pins "talk" PS/2. There must be a dozen different ways to make it work, and I suspect most of them had been tried by different mouse manufacturers over time. Knowing how any given mouse does this detection would require some investigation.
Will the device have the same performance as before?
Yes and no. Those that like to tweak their hardware for maximum gaming performance will prefer PS/2 over USB for reasons that can get complicated quickly so I won't go into details on that. For most any user the performance difference would be unnoticeable. PS/2 allows for better performance but that doesn't mean a mouse will work better with PS/2 than USB since there could be some other weak link in that chain than the port used to connect to the computer.
Is it necessary for the USB device to 'know' it is connected to a PS/2
port? I suppose it is necessary since USB-to-PS/2 adapters are just
that: adapters, not converters. They do not do anything to the signal,
so probably the keyboard or mouse should know and adapt their signal
accordingly.
And another question: Is there any latency added when connecting a USB
keyboard or mouse to a PS/2 port using an adapter? I suppose the
adapter itself does not add latency since it only connects some pins
to some others, but inside the mouse or keyboard something has to
change. The mouse has to switch to PS/2 data serialization instead of
USB. In theory, PS/2 should be faster, but I am thinking that if that
particular device is made to work natively on USB, it could have sort
of an internal signal converter which may generate some latency.
Without knowing specifics there's no telling which would give more latency. If someone is seeking the maximum performance from their mouse then that would be more on the selection of the mouse than on the decision to use the PS/2 adapter or not.
