I know when my G shepard bit a lady on the butt who walked into our house without permission and she filed a claim the insurance company would no longer insure the dog.. of course we owner occ so we kept Tessa loved that dog... was not the end of the world.  

Hindsight is great, but you shouldn't have let her move in without his written permission and there should have been a time limit of her stay (eg 30 days ) assuming she was in town to help care for him.

Broker says I cannot sue under agent’s E&O policy because it was supposed to be 15 days contingent even if she had not waived insurability without my permission.

See what is permissible in terms of the use of the property and set back requirements for structures.

Will they try to fill that vacancy on their own without asking permission?

Keep in mind that you will need everyone's permission to do anything on this property. 

Here's why I think we should raise the rent by $20-$70 a month:-Originally they moved in as a family of three (Father, mother, and small child) but they then moved in their mother without specifically asking for permission (they said she was going to stay awhile).

As far as I know you don't need a tenant's permission to report to the credit bureaus in the US, but I add a clause to my lease so they are aware. 

@Paul C.In general, Unless you want to build on the property line you don’t need permission from a neighbor but check your covenants if you have any.

Would you recommend moving in on my own first, and later request permission to add someone to the lease?