In medical emergencies so they really cut off the patient's clothes?

Yes, after trauma: Unless easily removed otherwise & especially if pt uncoscious.

Yes: In an emergency every second counts and it is often difficult to do any type of manipulation of the patient to remove the clothes.

Paramedics: And emts follow protocols. Treat airway , breathing and circulation. Then expose injuries. Without exposure, fractures, stab wounds, and gunshot holes are missed. That can be life threatening in of itself. So yes they do remove the clothing of trauma victims.