There are ring-tailed cats in North America. I know the miners used them for rodent control.
Not sure when they arrived.
Possibly not. The thing to remember is that while cats were likely attracted to humans by agricultural pests it doesn’t necessarily follow that they are or were ideal controls for agricultural pests. Basically they can only eat so much - you have to knock the infestation down first, then introduce cats as a control on what is left. So their impact to man may have been visible but not necessarily statistically significant depending on the situation - see here.
They make great pets, but how exactly do they reduce the consumption of grain? Why doesn’t the ratio of rats:cats end up in an equilibrium where the rats eat all the grain, and the cats eat enough of the rats that their population remains constant, and the cat population in turn is constant.
Only way I see it working is that you store the grain in concentrated areas. And the cats in turn guard it. And you just need a few cats to deter any rat from trying it. (rats can hear cats and see their kin dying)
And humans would need to feed the cats, but it’s costing them less grain than the rats would eat.
I suppose humans would need to underfeed the cats, to make sure that they kill the rats. Though that might not be necessary - cats will kill animals just for fun.
You’ll never get rid of all of the rodents completely. They will, as you say, come to some nonzero equilibrium amount. But the equilibrium amount of rodents is much lower, and the equilibrium amount of grain much higher, with cats than without.
Ring-tailed “cats” are not true cats but are a related to raccoons.
Would lynx and bobcats prey upon agricultural pests and rodents in a way useful to farmers, or would they just eat your livestock?
Unless you count dogs, North American indigenes didn’t have livestock.
Depends on how well you domesticate them. Dogs (aka, wolves) seem to lay off the livestock.
They had turkeys and ducks as well (at least, the Aztec did)
That’s because the ones who attack livestock are killed.
The Aztecs are generally considered Mesoamerican natives. North American natives (or if you insist, Indians north of Mexico) had no domesticated animals other than dogs until various animals such as the horse were introduced by Europeans.
Well, we’ve seen some references in this thread to North American grain storage, in claw pots. And its been mentioned that north American rodent species aren’t as bad as the European rats and mice. However, how the Mesoamericans kept the massive amounts of grain that their large stone cities would need, is also interesting to me. They’re at least jungle adjacent, so serious vermin infestations could be a problem.
Unless I’m completely mixing up location, city size, and vermin species. Which is likely.
The Aztecs apparently stored large quantities of maize and beans in large binsmade of wood and mortar. The article doesn’t mention how rodents were controlled.
The Aztecs at least weren’t “jungle adjacent,” living in a relatively dry climate.
Pots made of cat claws, obviously. 
When I was in grade school a world history book had a cute story imagining when cave kids talked cave parents into keeping cave kitten/cub …
If you went by that I wonder on a daily basis what if cave human just ate the little monsters instead of making friends I mean itd save the couch curtains rugs ect and hundreds of dollars for cat food litter ect
But from what ive read native americans weren’t sure on what to make of the cats the Europeans brought over but they really never kept the dog packs as what we consider pets either they used them for work or occasionaly as meat …
Well yeah, its already been established, just having cats around would repel vermin. Why not keep a few fetishs if its all the same.
They called the suddenly appearing stinging insect known as the honeybee the “white man’s fly”. They had the peculiar tendency to appear along with European settlements.