Thank you for choosing Just Answer. I would like to assist you with your question today.
Your landlord cannot charge more than the actual cost of repair/replacement for the parts of your unit that are damaged or worn beyond the expected normal "wear and tear" of the unit.
This means that they can only charge for the carpet replacement of the entire room if it is impossible to replace the damaged section alone (they have a right for it to look uniform, but if there is a less expensive option, they must choose it).
They cannot charge a fee for a service dog.
(Even if they could charge a pet fee for a pet, they cannot do so at the end of the lease unless it was disclosed in writing at the beginning of the lease).
Under no circumstances can a landlord keep you from your belongings. Landlords cannot use "lock outs" to force payment of rent or other fees. They must make your belongings available to you promptly (and they must keep them secure, they cannot just dump them on the sidewalk and tell you they are available).
Your landlord is at risk of significant penalties if they do not provide you with your belongings promptly, hopefully they will provide them to you. While the dispute over the cost for repairing the carpet may not be as easily dealt with (it comes down to how much carpet really does need to be replaced, and how much just needed to be cleaned, but a round number of $1,300.00 usually means they are charging a set amount rather than the actual cost), I do hope that you do not have to file a claim with the small claims court, as it is somewhat time consuming, but the Courts will enforce your rights. Keep track of all of the phone calls and communications you have with the management as it may be necessary to have these notes in the future.