Are there any insurance policies that cover damages caused by stray animals? If not, should there be?

No, vermin is generally an exclusion on most homeowner/landlord policies. I think the reason is it could be prevented with some basic homeowner effort. I just had (2 weeks ago) 2 mice get in my house and trapped too. Got some caulk when around the basement exterior wall and caulked all holes I think they could get through…no more mice! Using preventative measures could stop any damage.