Is it a bad idea to pour water on a pan without letting it cool off?
Depends on the pan and how hot the pan is ; glass pan is likely a problem, and some of the glass-like decorative coatings on metal pans are a risk. Otherwise temperatures would need to be well above cooking temperatures to even consider damage to a metal pan. Hot enough that the water will "bounce" out of the pan because of rapid steam development.  Aluminum could be a problem at 800 F ( Melting begins around 1000 F ). Magnetic stainless ( 13 % Cr)  is happy as auto exhaust to 1400 F ( red). Non-magnetic stainless ( 18-8 ) will go higher. Cast-iron and copper can go 1300 F + ( dim red) : however the tin coating on older copper will be destroyed above 600 F... BUT , some pans will warp even at normal cooking temps; I have had a couple inexpensive aluminum frying pans develop a rounded bottom so they won't sit flat on the stove (no water cooling involved)