A tree may be big, or tall, or have long heavy branches, but still be safe - trees adapt by growing strong enough to take the forces on them. Many trees hollow out over time quite naturally, and may even be safer hollow than solid. Call in a tree consultant only if a tree moves right down to the ground, or if the ground moves. Ivy can increase the windload, but it gives shelter and food for insects and birds. Only cut it back if your garden is especially exposed or if it is growing over leaves.
All trees drop dead leaves.
Some trees naturally shed bark, and most trees will shed twigs and small branches as part of their natural pruning, when other branches shade them out. You can turn all of these into leafmould and compost. If big branches fall, there may be something wrong - get professional advice. Ask your insurance company if your cover includes trees. A tree will regrow after it is pruned.