For methane produced by farming all components remain in a cycle. Everything released by animals eventually finds its way back to plants, which get eaten, which become exhaust of animals, etc.
This sounds to me more like a spin by big oil to finger point at another industry.
To be exact, excess methane go in the high troposphere, where it is broken down in CO2+H2O (the half life is very short, I don't remember which is which but it's either 5 or 20 years). So in any case, it accelerate climate change.
Now dairy farming? I would not be surprised if they feed mostly all grain and beans. Every part of dairy farming is about industrializing both the process and the cow with a lot of selective breeding for ridiculous amounts of milk production. Milking cows now produce over twice as much milk per cow than they did in the 60s and certainly require more nutrients to maintain than a natural cow's diet would sustain. Some cows are producing 10 gallons of milk per day now.
No, they don't. Methane is not absorbed by plants and is not part of any cycle. The only reason it is not a catastrophe is that it breaks down in a relatively short time. If we stopped producing methane in this way, no other process would replace it and the planet would cool down.
The obsession with CO2, specifically emitted by fossil fuel use, for me indicates geopolitical motivations (for both standpoints).