What if the pet causes damages to the property ...

You are Responsible. Legally but not only that but why do you think you wouldn't be responsible morally?  I know I would try to fix anything I tore up that either I, or my pet or my child tore up? Don't you think that is right.  Think about it if you owned the property and I trashed it. 

your responsible for all damages if its your pet and you are renting they could take it out of your security deposit or make you pay for it out of your pocket..

you are responsible for any damages that has been done

Everybody is of course resposible for what their pets damage.When my pet cows got out and "danced"on a neighbors lawn and ate his plants my insurance company paid because I could prove they were pets so ring up your insurance company if they pay out for cows they should for cats or dogs but if they had not paid out for the damage my cows did I would of course have paid to have his garden put right and you should do the same.

What do your lease say? Most of the time you are responsible for any damage caused by your pets or children or even yourself or anyone else residing in the residence. If you rent from someone it is morally and lawfully your responsibility to repair all damages or pay for damages incurred. I't just right.

You would be responsible. It is unlikely that any home or pet insurance company would cover for this.