Can I use some of the tenant's damage deposit to pay what my insurance does not pay? 

A good operator will have insurance that protects themselves and you as the landlord from damages and liability.Is your property in the OC?

Insurance companies, and this includes umbrella policies, will deny or challenge your claim especially if the damage / claim is high and based on negligence or any fault of your's, READ your small print and don't think you are so good at landlording that accidents won't happen.

The vast amount of damage is normal wear and tear.

I’ve been with the same insurance company for 5 years and never had a claim and likely won’t make a claim unless I sustain a large amount of damage somewhere.

In general, when the landlord's property causes damage to tenant's property, the landlord is liable, unless there is proof that the cause of the water leak was the tenant (unlikely I presume).

@Malick NgomGet an attorney to send a certified letter requiring signature and include the satisfactionIn most states a lender can be held liable for damages for not recording the satisfaction

It's hard enough when a tenant damages my rental and it happens. 

I also don't think it's common to convince the appraisal district that your property did not appreciate at all, unless you can back it up with pictures that your property was damaged or is in worse shape that it was the year before.

Early on in my investing journey I experienced a ton of water damage because of water leaks in places that often goes unnoticed under sinks, in utility closets etc.I was curious if anyone else here has considered any remote monitoring water leak systems?