If you look at PCF8575.h, you will see that the constants P5 and P6 are defined as 5 and 6 so the NewPing library will be using the Uno pins 5 and 6 to attempt sonar, not pin 5 and 6 on your port expander.
As @jremington says, you can't use expansion pins for sonar.
In general, you can't pass expansion pin numbers to any function expecting Arduino pin numbers.
Use the expansion for pins which you do nothing to but pinMode(), digitalRead(), or digitalWrite(). Those will work by changing the calls to pcf8575.pinMode(), pcf8575.digitalRead(), and pcf8575.digitalWrite(). Hopefully, that will free up two Arduino pins you can use for Trigger and Echo.
Does the pcf8575 library support other Arduino functions? If there is a pcf8575.pulseIn() you could use that to read the Echo pin.