Servos, commanded using the Servo library are sent new position data based on timers. There are limited number of timers available. Those same timers are responsible for PWM on the PWM pins. Using a pin for a servo effectively removes the ability to perform PWM on some pins. Which ones depends on which timers has been usurped for the Servo libraries use. Which timer is usurped depends on the pin the servo is connected to.
Timers trigger interrupts when the timer ticks. How that would affect the pulseIn() is unclear.
Ping sensors do not return a value to an analog pin. Some other kinds of ultrasonic sensors do. You realyl would need to tell us what kind of sensor you really have.
You include the NewPing header file, but never instance the NewPing class. Including the NewPing header file is, therefore, useless. Why do you do it?