The amount of damages owed is governed by the written contract or by the court's determination of 'reasonableness.' There is no question that you are liable for the damage, but as to who, when, how, the carpet is replaced and timing and other pets.....these are factors that are covered by the rental contract or will be determined most likely by a small claims commissioner/judge. 'Health and safety' in residential housing are HOT SPOT type words that cover many sins and are emotionally loaded.
I'm sorry you're in the difficult position you are in. It appears you love pets as much as I do.
First, talk with your landlord and try to come to an understanding. Take the high road and admit you breached the lease, but appeal to her sense of humanity since you were trying to do a good thing. If the cats caused damage, you will be responsible for it, so you shouldn't fight it.
If your landlord won't work with you on this issue, BEFORE YOU PAY ANYTHING, make your landlord/owner provide you with a formal written statement to you that includes the following: (1) why her course of action deviated from what she told you she was going to do; (2) what percentage of the damage she determined was caused by your dogs (on the lease and approved) and what percentage of the damage she determined was caused by the cats (in breach of lease agreement); (3) copies of the written quotes she gathered to determine whom to hire to inspect; (4) copies of the written quotations she gathered to determine whom to hire and what carpet to use when she replaced the carpet; and (5) copies of any other documents related to what she is billing you for.
If your landlord won't provide any of it, you may have good cause not to pay for the new carpet.
Unfortunately, the cats have to go. You are in breach of lease agreement and, as such, you waive several of your rights as a renter when you breach the lease. Next time, don't hesitate to talk with your landlord in advance of anything reasonable that you want to do that is not allowed in your lease. Normally, landlords would rather have you stay in your property and pay rent, than have to move out - forcing them to find a new renter.
To determine the scope of your rights, one would need to see the contract governing your relationship with the tennant. Nevertheless, I can give you some hints:
1) From the facts presented, it seems that you and landlord are both right to a certain extent. You may need to pay part of the amount sought by the landlord.
2) You should first try to negotiate with the lendlord before you hire attorney and make further attorney related expenditures. This could cost more.
3) It appears that you were not allowed to bring in the two cats because you paid fees related to dogs but no cat fees. This implies a breach of contract made by you, even if the cats were brought in temporarily. It is possible that you were obliged to obtain landlord's permit to bring in the cats and if permit was granted, to pay a cat fee. It appears that you did not do this. Still, you may have several good arguments to substantially decrease your liability and the amount sought by the landlord.
4) It is important that the landlord's "experts" said that the damage was primarily caused by dogs. It would be good for you to get this statement in writing. This statement suggests that the carpet damage and replacement was not primarily due to the cats but due to the dogs that you kept legitimately in the flat. This fact should substantially mitigate your liability because it shows that the carpet replacement was not needed in relationship to the "illegitimate" cats.
5) Look in the contract to see what was monthly fee of 25 USD fee payable for. If the fee was payable, inter alia, in exchange for usual tear and wear (ideally associated with dogs in the flat), then you may argue before the landlord, that the amount of the fee he is asking should be substantially reduced, because the fee could be deemed to allow the landlord to replace the carpert on a regular (e.g. annual basis) after is used up (exposed to usual tear and wear).
5) You should ask the landlord about the price of new carpet and the price of services associated with the replacement. This will allow you to find out more easily if the amoun sought is reasonable and ordinary. I am not sure how big the flat is but 1400 USD may be too much for simple carpet replacement. You may want to check what are ordinary costs of carpet replacement (choose another well known average company) doing the same service. Make sure you tell the company what type of carpet was in your falt originally in order to calculate their offer. Make sure you obtain the offer by e-mail or in writing so that you can present it to the landlord.
5) If the offer is cheaper than 1400 USD, you may present the cheaper offer from another company to the landlord. You may then argue that the costs the landlord is seeking (1400 USD) are not ordinary. This could allow you to further decrease the requested sum and further limit your liability.
6) Please note that the landlord should not be allowed to bring in a new/more expansive carpet than the original one and charge you for the cost increase. I doubt that the contract allows the landlord to do this. If this was permitted by the contract, it would require your consent. In case the landlord replaced the old carpet by a new one of the same type, see point 5 above.
7) It is questionable whether or not the landlord had a right to enter the flat without your permission and exchange the carpet. This would depend on the contract and the circumstances. In principle, the following rules apply: In case the damage was not serious and there was no risk of contamination or damage increase, there is a better chance that the landlord breached your right and entered the premise without a legitimate reason. This should reduce your liability. However, in case the carpet damage was serious and subsequent use of the carpet would cause further damage or spread contamination, then the landlord was probably authorized to enter even without your permission.
8) Best approach seems to settle with the landlord