Hiya, The green wheely bin could be dealt with by a coating of vasaline or grease of some sort around the rim, so no ice could get a grip. Might get dirty, but it's worth a try.
The woody cuttings are great as kindling on my friend's lounge fireplace, and the kids do like to sit and feed the flames! She kept them in a bag in the garage and let them dry out.
I have a wormary (in my greenhouse, to help the worms survive) and they are still munching, although slowly, as I have a heater in the greenhouse for the plants.
Our compost heaps are regularly raided by our neighbours' cats and the local squirrels as there is always a range of food goodies available to eat. I don't know which animal carries the empty egg shells away, but I find them around the garden!!