Generally speaking, drastic fluctuation in temperature can make rabbits ill, so regular excursions from the warm indoors to the freezing outdoors doesn’t seem wise; however, if you want to give your rabbits some stimulation, I’d say 5min or so while heavily supervised by you and with appropriate safety measures (no chance of escape, mindful of swooping hawks, depth of snow and presence of hidden sharp branches/bushes) can be an enjoyable experience for some rabbits. It will depend on your rabbit.
A lot of rabbits on social media will be very cute, on leashes, hopping in the snow and such, but I imagine those rabbits are either 1) regularly acclimated to being outdoors and/or 2) taken inside shortly after filming for social media purposes.
Their coat does a good job at making them stay warm.
For your bonus question, it seems that will just be a behavior that may stick with him. Wick still does territorial marking on the sofa when he’s lazy or when he goes somewhere he’s typically guarded from, so a neuter is not a cure all.
The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.