I am currently sub-leasing an apt and found out the original leasor (my landlord) that I rent from is being evicted. The apt owner is not kicking me out and I will have to sign a lease with them, plus security and pet deposits. How do I get the refundable security and pet deposit back from the original lease? Can I request proof that the deposit was held in it's own account? What are the laws regarding this?

Speak to the owner of the apartment and find out if they are already holding a security deposit from the person who originally rented the apartment. If so then they may be willing to credit you with that deposit.

If they are not already holding a deposit then unfortunately you will have to pay a new deposit to the owner and then go after the person who originally leased the apartment to you for the return of your security.

The apt owner is using their deposit to pay the back-rent. I don't think they ever paid the pet deposit. Most likely I will have to take them to court. Can I request proof of them holding my security deposit in it's own account?

Are you talking about proof that the person who originally leased to you held it in a separate account? That doesn't really matter now, does it? Either they will pay it back to you (unlikely) or you will have to sue them for it.

If you're talking about the apt. owner holding the new security deposit in a separate account, sure, you can ask for proof. But if the law in your state doesn't require that they do so then they don't have to provide it.

Not all states require that security deposits be held separately. I couldn't find and law in SC that required that.

Also I am confused, is this a deposit that you gave the other tenant as "YOUR" deposit

OR are you talking about your previous landlord, you have no claim to their deposit if you were only on a sublease