I was thinking the exact same thing. from the size of the ice hanging from your roof if that is from this year. Insulation is more of a consern than the leaking roof. I am guessing from the leaky skylight that your place has cathedral ceilings which are one of the most difficult attic areas to insulate properly. I dont know how old your roof is. but im guessing from the info given the lack of circulation in the attic is causing the leaks.
Yes you can put a roof on inwinter. no ashpalt shingles wont bond as good and wind blow offs can be a concern. I would suggest a steel roof in the case. because they are great to stop ice from damming up. and they dont use heat to bond to the roof. they are all screwed down…
my consern for a steel roof from that picture is the avalance like snow slides that can occur with a steel roof right over your driveway like the picture shows.
My neighbor has a steel roof on the 3 story house and the two HUGE snowfalls we had last winter, october and Superbowl sunday last year… the over2’ of snow came crashing down off of the roof a few days after each storm. luckily his cars are parked enough away from his house that nothing and noone was crushed.