Yes, winter is a great time for pruning and removals.
Especially removals, snow and ice can be the main cause of trees falling over, splitting, or losing limbs.
Yes, you are fully liable. Do not wait for an insurance claim, get the work done as soon as you notice it is a hazard.
If you have a tree that is growing near any kind of structure and you are worried about a large limb falling, you can install a brace to help reduce stress to the tree.