As we all know, some of us live in the white states (aka snowy states). We all have been stuck in the snow before.
I've been stuck only twice in the two years I've owned my car, I was trying to go up a hill but the tires were spinning alot. My dad later told me not to give it too much gas or keep it spinning, it'll hurt the transmission.

My question is, does transmission fluid go bad fast or get on its way to bad during those situations?

the most common killler of the Trans fluid is heat

If you are just spinning one wheel, then yes you could be harming the Trans

Yes, its the heat that will kill it.

But remember, if you are spinning your wheels....you aren't really going anywhere. Wheels slowly spinning on ice is going to have you moving just as fast as WOT on ice.

The wheel spin will not hurt the transmission. What a lot of people do when the get stuck in the snow is "Rock the Car". That is, shifting from Drive to Reverse and back to Drive repeatedly until they Rock the car out of snow patch they dug themselves into. THAT will put a strain on the transmission. But the transmission oil will not "go bad" during the winter. But it wouldn't hurt to check the condition of the oil by getting the engine up to operating temperature and checking the level and color of the oil. Bright Red is good. Dark Red it'* time to change the xmission oil. Black? Well it might be time to save for a rebuilt xmission.