It doesn't sound like there was any damage to the property when you viewed it. Yep they are being suss by not allowing you into the room. I would trust your instinct with that, they probably are hiding a pet. However, your stress seems a bit of an over reaction to me - I hope you don't mind me saying that, I'm not trying to be condescending - but I've been a landlord also and the issues you've mentioned don't add up to big problems like property damage or default on the rent. I agree with just breach for what you can - the lawn maintenance - then basically let it be until the lease runs out and advertise for new tenants. Backpacks left out, and ironing board - that is not a sign of a bad tenant. It's just people living.
Also if you are attending inspections yourself and investing so much time into the management of the property, have you considered saving the 10% that agents pocket and just managing it yourself? That was honestly the best decision we ever made with property management (well, that and to sell it and never be landlords again ).