Inherited Tenants and the lack of leases

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I recently closed on a house with some inherited tenants.

Both have pets with no pet addendums or security deposits or increases in rents for these pets. 

One of these tenants is breaking the counties rules on # of pets allowed.

-I am up for keeping them especially in this time we are in but wanted to get some peoples input on this.

Can I retroactively charge security deposits moving forward for the pets that currently live with these tenants... i.e. they are re-signing a new lease with me I'm going to tack this on. ( I think I answered my own question here)

-I was also going to inform them of the state statute and if they don't comply give them a notice to vacate the premises or contact the county for an inspection of the premises.

Second, 

Both the tenants have stuff strung out through the side yards as part of an agreement with the old landlord. 

Apparent they have agreements to utilize things not in their lease.(the shed, closets, etc)

They currently use these things but If they want to continue to do that moving forward i think i am going to charge them or have them move their stuff 

Has anyone ever dealt with this?