To rent, apartment owners do not have to paint walls and replace carpets by law before new tenants move in. To rent, the apartment must be habitable by law.
Pet urine and pet markings on walls and floors can make an apartment uninhabitable due to smell, sticky odor residue and general health reasons. So, it is usual for such damage to be taken care of after the tenant moves out and before the new tenant moves in.
The real problem here is that you don't have a check-in or check-out punchlist that you and your landlord signed. A pet deposit is one thing. But simply having children and pets per se is no reason to replace carpeting after 2 years -- a good shampooing should be enough depending on the kind of pooch and whether the kids spilled a lot of things assuming there are no poop .... stains and smells and yes that's what the deposit should be for. Painting? Every place needs a touch-up too. But some landlords are pickier than others.
But the real problem is, even assuming you're 100% right, all you can do at this point is fight about it and that will probably cost you more time, effort and money to do so, or you can remember in the future to have the place checked out when you move in and leave, do some touch-up yourself (take pictures), keep receipts.... better you control the move-out costs (you can even work with the landlord at the end to apply the pet deposit toward a good shampooing if the landlord is reasonable -- maybe a big assumption) than the landlord. If a landlord won't do a check-in punchlist, make sure you do a fairly good inspection as soon as possible after move-in and notify the landlord asap (in kind tones.....). And while check-in/out documentation isn't a guaranty, at least it gives the landlord something to think about before redecorating on your dime....
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