Responsible for dog bite?

....good advice here and do not put anything in writing with this person until you speak with your insurance company , let them handle it. Stop speaking to the person holding you responsible, defer her questions to your insurance company, thats why you  pay insurance. I bet you didnt even know the dog would bite.,right?
Good luck to  you....

Your insurance company is the one that should advise you on this.  All property owners have tort-liability for injuries that occur on their property.  Renter's insurance is entirely different than the liability insurance that an owner usually carries.

You need to read the liability section of your policy to see what it states it covers.  If there is a tort condition on your property, you are liable, which is why you have liability insurance.

As far as the vandalism of your home, that may or may not be covered by your policy.  If you only had insurance as "owner occupied", then tenant vandalism will not be covered and that is your own financial issue.

Similarly, a tort condition created by a tenant on your property may not be covered by your liability insurance if you have insured it as owner occupied.  A lawyer will probably not be of much help in that condition.

Start with reading your policy, then calling your insurance agent to see how they interpret what is covered.  Then see what you can work out with the injured neighbor.  The neighbor will remain your neighbor unless you sell, so you really need to try to stay on good terms with them.

Unfortunately, most people that get into the rental business have no clue what they are getting into.  A lot of that is due to false propaganda spread by NAR and their members.

Contact an attorney and your insurance company before doing ANYTHING else!  You're dealing with a potentially expensive and explosive situation, so do NOT take the advice of folks on a public real estate forum like this.  Seek legal counsel immediately!