That sounds like an excessive amount of money for carpet and a vinyl floor. Are there other charges besides replacing the flooring?
Sure, you can ask for proof of when the carpet was installed. That doesn't mean that they have to provide it or that they are even able to. Often a dispute over rent, a security deposit, or damage to an apartment is controlled by terms of the lease. Does your lease speak to the carpet and vinyl floors? Did you pay a security deposit? If so that may be applied to the damages.
If you want to dispute charges that you landlord is claiming the best thing to do is take your lease, any photos of your apartment that illustrates the damage (or ideally lack there of) to an attorney who can help you determine what your liabilities are before you pay out such a large sum.
DISCLAIMER: this is not intended to be specific legal advice and should not be relied upon as such. No attorney-client relationship is formed on the basis of this posting. The law firm of Klish and Eldreth, PLLC offers many flat-rate legal services in various areas of law in North Carolina. Your inquires are always welcome. Feel free to call 919-833-5322 for a free phone consultation or email Justin@klisheldreth.com