Powering 3 servo motors and Arduino Uno with one 9 V battery
I don't think anything should get damaged; the regulator is rated for 1A, and even beefy servos don't draw more than 200mA when moving against heavy load. The problem is that your battery won't last long. Even the heavy-duty Li primary 9V batteries are under 1000mAh, and the actual capacity is heavily dependent on the discharge rate. 

Why can't you use a rechargeable Li-ion pack with an appropriate step-up or step-down converter?