Snow? Errrr if it ever snowed here the whole town would shut down. No gig then! Having never driven in snow, if it did snow here, I'd have a hard time getting there and might call it off...not worth the accident risk....
In my area,
If it's an outdoor gig and it's raining, call the venue. 90% of the time they cancel and are grateful you asked.
If already there and it starts, depends on the amount of cover....and the amount of rain....etc. I keep garbage bags in the gig kit, speakers work with bags over them. Amps have tarps, etc...but that's only for light rain.
If it starts lightning, I'm done. Not playing in the lightning, it's not worth the risk. That's a gig ender for me.
Heat? Depends-if there's a cover so not in direct broiling sun, sure....I've played gigs where I sweat so much the strings just slide out from under the fingers. No cover, no band...did do a one hour set without a cover once, but it was spring and not broiling. Honestly, haven't had this be an issue otherwise, most places around here understand the band needs a shade.