Hello Donald,
Wow, that's quite a lot to take in. Let me see what I can do to provide some guidance. I am sure I will miss something, and if you like the responses I give, feel free to hedge further inquiries about what I omitted.
You could always consider suing the property management company for breach of contract. Assuming, of course, you had a contract with them stating clearly that they needed to screen and accept tenants according to specific criteria, etc. But that is a costly and complex process. Oftentimes you're better off taking the loss there.
What I recommend: establish your authority with the tenants. As you have mentioned, they are giving you some push back. That is because they've had such success in doing that with the property management company. Granted, you don't want to be rude or abrasive in your undertaking -- you should be courteous but firm. It's not their fault that the previous PM didn't follow the screening criteria, but they are in your rental, not their own home, and with that comes conditions.
Explain the situation to them: You are here because the property management company didn't manage the property to your expectations. These inspections should have been occurring, but they haven't. (Because of that, things have gone into disrepair. Your presence here is solely for their benefit.) Rent payments have been lenient, but now they aren't. From this point forward, you (the tenant) will be expected to abide by the contract to which you agreed when you moved in. You can even let them know that you are willing to let the past issues slide. But from this point forward, you pay rent on time or incur a late fee. If you do not pay the late fee along with the rent, you receive notices, then get evicted if you still think you don't need to pay.
You could even go so far as to tell them how things are going to be and if they don't like or agree with that, then offer them the option to break the contract and move out.
As to the pets, it's tough to impose fines if the contract doesn't say anything about it. I can't elaborate on it too much unless I see the lease, but you said you have a clause about prior approval. I wouldn't recommend letting much anything "slide" at this point. To establish authority, you need to bring everything up to par on your lease. They didn't follow the approval steps, even if the dog is outside. Offer them the opportunity to "approve" the pet. If your lease says anything about a pet fee, tell them that they need to satisfy that.
This is definitely a tough situation because you don't want to come in looking like the bad guy and then have these tenants damage your property out of anger. If their credit scores are that low, chances aren't high that they'll care if you threaten to report them to the credit bureau. You could always offer them the "cash for keys" option. Tell them you'll give them three hundred bucks to move. They just have to tidy your place up.
If I missed anything, feel free to reach out. But to reiterate, they signed a contract, they haven't been following that contract, it's been okay up until now, but now you (the tenant) will follow the contract just as I (the landlord/owner) will. You (the tenant) are contractually bound to pay the rent by the first of the month and I (the landlord/owner) am contractually bound to assess a late fee if you don't pay it within the legally permissible time frame.