If something like a roof tile flies off my house in the storm and causes damage or injury, am I covered for my liability?

Your buildings policy will usually cover your legal liability in instances like this. There may be a limit to it depending on the policy.  For anyone to be able to claim against you it would have to be shown that your negligence led to the damage (for example, if it can be shown that you were aware that your roof was in a dangerous condition, and you did nothing about it).