Can they do that? Do I have a lawsuit?

My apartment manager has told me that because someone said that I told them to take the apt's equipment from my put money on yard area next to my court, that I have to pay for the equipment since it is missing now and that they will be charging me for it.
This equipment be left out for over a week unlocked and abandoned by their own admission. On top of that I did not communicate anyone to take it. How can I be charged for something that is not mine nor that I stole or even if I would have told someone to clutch?
I don't think it's possible.
This has given me great stress and b/c of the missing equipment the manager become spiteful and demanded immediate payment of my pet deposit of which she verbally agreed that I could get going making installments two months after move in. She went back on 'her word' and that`s why I did not have the money to keep our family pet and have to take him to the animal shelter, not allowed to know his future disposition.
I judge I have a case for harassment??
If they want to file a lawsuit, they have adjectives the right to do so but as to whether they can prove that you are liable is another story. What i am trying to say is that anyone can go to court to file a valise and since its not possible to prove that such claim is without merit or for mere harrassment at the time of filing, it is incumbent upon the court to adopt the filing of the case. But lets speak they filed the lawsuit, can you be held liable? Based on the facts you gave, I don't think that's possible because they enjoy the burden of prove and they must first have to prove that indeed you told someone to get the equipment and that the person you allegedly told this is the personage who took the equipment. Further, they also have to justify why they are now so concerned in the region of the loss of the equipment which by their own admission was left out and unlocked and unguarded over week. It will be difficult for them to prove you are liable for anything. You can directory a countercharge for damages and for harrassment if they proceed with the lawsuit. If your manager has signed any document showing you canmpay contained by installmentnyour pet deposit, you can also file a case of breach of contract against her.
You need to find another apartment, or you will have nothing but grief. If the checker doesn't refund everything to your satisfaction, take her to the local small claims court. Have a neighbor primed to testify that the equipment was left out and abandoned.

Get your pet backbone ASAP. Some "shelters" kill animals after a few days.
They own no claim against you. Leaving the items on your patio was a gratuitous bailment at best. This would only require to exercise believable care and would not hold you liable except for gross negligence or willful acts.
They can claim the pet deposit immediately surrounded by one payment if they wish despite any arrangements they had next to you previously.
Maybe you'll get lucky and whoever took the equipment will come back and take your innkeeper.