After removing the winter dieback, I prune by how much I want to shape a bush and if the bush needs to be cut back. I judge each as I come to it. Severely pruning hybrid teas and grandifloras normally will produce longer thicker canes and larger blooms. Lightly pruning will produce more but smaller flowers on shorter thinner canes. Floribundas and shrubs where you want many blooms can be lightly pruned. If the bush has many older thick canes, bare bottoms, and needs to be rejuvinated, cut them way back.
Austins are shrubs so prune them as you would a shrub. Most any climber in zone 4 will need to be cut back quite a bit to remove dead canes. I'm in 5a and except for three of mine and Seven Sisters, a once bloomer, I have no choice but to cut them back. Once blooming climbers, except for damaged and dead canes, prune after they bloom.
If winter's kind to your climbers post again but I doubt you'll have much choice in zone 4. Aside from New Dawn, Quadra, and Ramblin Red, my climbers need to be cut back to 3 feet or less, usually less and often to 4-6 inches.

In late October when our yard man and lady came to do the prunning and mulching, the rose bush was still with many blooms so he chose not to prune it back. It's winter now, can I do that myself without causing damage to the rose bush? Can I take a cutting from it now and try to root it? I was thinking that if I cut it back now, it might be exposing it to too much cold but I have lots of dried leaves that I could put on top as mulch. Even had some extra mulch left in a tub in garage that I could place around it. Seems like all the leaves from the neighbors end up in our breezeway in one corner. Or should I just leave everything alone until Spring and warmer weather?

Why do you want to cut back now? Leave them alone until spring. There's no benefit to the rose bush to cut it back in the fall. Too often we do what benefits us with no regard what's good for the plant.
Nature has spent all summer storing food and moisture in the canes and roots to help next springs leaf out and early growth. Those first leaves you see are the result of overwintered food and moisture.
Prune in spring to remove winter dieback and damaged canes.

That's the thinking of the man that does our prunning. He will do it in early Spring for us. Do you think that I can take a cutting now or just wait to do everything in the Spring? One of the reasons I wanted to cut it back now is that the canes are brushing up again the window screen when the wind blows.

Cut back only enough to clear the window screen then do the rest when it's time.
If the canes are not dead, you can try rooting them but cuttings from actively growing canes root best. I have rooted cuttings that were overwintered in the ground. They were cut and buried the previous fall.
I take my cuttings right after the spring flush and up to the end of July.