But I'm concerned about liability if someone gets injured.

Who is responsible if there is any damage to the house or if anybody gets injured in the property and sue the owner of the house during the time from now until the certificate of redemption is issued?

The fire was contained and luckily no one was injured.

In my opinion, if you have a tenant injured or worse by CO there isn't a jury on earth that would not find you negligent in civil or criminal court.

If you provide extinguishers you are obligated to provide proper training on it's use and in the event of a fire, where a tenant is injured attempting to contain a fire, you can be held liable.Better pray the extinguisher does not malfunction or the tenant does not use it improperly resulting in injury or death.

What happens if 1 of those 10,000 appointments got injured on site while they are doing their job, can't the department be sued?

The ex can stall things and just try to injure her financially and economically which is what she is trying to avoid. 

even if it does or doesn't, wouldn't it make sense to have all the liability on a company just in case there is an incident where someone is injured or worse and insurance wont cover?

No one was injured by their negligence, no water damage was done, no personal property was lost.Most places require renter's insurance for the liability aspect such as dog bites or the pizza guy hurting himself on the premise and wants to sue. 

For example: Someone decides to sue me because I injured them during a flag football game (this is completely fictitious FYI).In this scenario, my worry is that someone might try to sue me, realize that I own multiple real estate properties, and try to essentially drain all of my money and force me to sell said properties.What is the best way to mitigate this?