Landlord Has Question About Lease

In May Lease I Have A Clause That I Added And It States That If For Any Reason That Rent Isn't Paid Within 30 Days And There Is No Contact By Tenant To The Landlord That Eviction Procedings Will Start And That They Waive Their Rights And Agree To A Speedy Eviction.
Now The House Is In New York Can It Still Be Disputed And Go Through The Courts And How Long Before She Is Actually Removed From The Property?
In Addition I Have A New Property And The Tenant Signed A Year Lease With The Agreement That She Can Keep Her Dog. My Insurance Company Did A Inspection Saw The Dog And Is Going To Drop Me. I Am Finding This With Many Insurance Companies. Nothing In The Lease Says The Animal Can Stay It Was Verbal. Can I Evict Her And Break The Lease Because I Will Not Have Any Insurance Coverage?