No, and not even all keyboards and mice. It will only work for ones which can do PS/2 signaling over the USB port.

Probably not, if we're speaking about the next adapter in the line. They both use 4 pins, and there are no active conversion in the adapter itself, it just connects the pins like this:

USB ↔ PS/2
+5V ↔ +5V
D-  ↔ D
D+  ↔ CLK
GND ↔ GND

So, as long as next adapter is not doing something funny with PS/2 signals, it should be ok for bare USB 2.0 connection.