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It's not quite the same nor remotely as common as with dogs, but yes, cats do sometimes hump at random things like inanimate objects.
It's more common with intact males or with males that were neutered later in life, but any cat including females might show the behavior.
However, as I said, it doesn't present quite the same as with dogs. With dogs, it can ...
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There might be more behind this, so in either case trust your vet more than me as a random stranger on the net.
It's totally possible the vomiting is connected to normal digestion/instinctive behavior:
Assuming the vomit is some color between yellow and light brown and mostly consists of liquid and foam with potentially a few small bits of leftover food or ...
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