Seems like you have several problems all rolled up into one.
1 - The kids at school. Like Tina said, how about Social Services? But think about covering yourself. Call the school (or schools) ask if they can offer any help with transport. If it’s a NO, then ask if one of the staff can drop off some work for the kids to do. Get on to Social Services - ask if they can help - “you have already asked the school to send some work home” - if they cannot help, ask who they suggest. Make sure that you have a record of day, time, person you spoke to, what they said. If you can prise an e-mail address out of them, send a confirmation message.
2 - Walking in the snow. I don’t use crutches, so what I say is biased. I use two walking sticks. I have learned to make sure that whichever stick is going to take my weight next is upright. If I have to let go of one, the other must be a support. If I have to let go of both, then I am leaning against something. And, on snow and ice, move very slowly.
3 - The car. Not enough data. Driving in snow is easy - yes, honestly. Driving an automatic in snow is even easier - trust me, I have done a lot of it. And that is in real snow - several feet deep, at times.
There is just one big trick - do everything a lot more gently and a lot earlier than you normally would. There is a bit more to it than that, but gentle and early always works. My old trick when snow came down was to find an empty car park and find out what happened when I pressed the throttle down gently, or hard, and the same thing for the brakes. Simple, now you have some experience (and more than a whole lot of people).
De-icing - the dear old black bag is your friend, here. Cut one or more up into the largest pieces you can manage. One at a time, put them on front and rear screens and side windows. Smooth them out flat (you are probably leaning on the car to do this) and air pressure will keep them there. In the morning, peel them off and the white stuff comes away as well. If you have a plastic bag hung over the hand-grip of one crutch, you can drop the wet bits of plastic into it (and dry them off later).
And, like the others have already said - do not go out unless you have to. If you fall, what will happen to the kids then? Stay safe fore their sakes as well as your own.