(Update: In hindsight I'm not sure you were talking about methane emissions in general or about methane from meat production. Here, I'm focussing on the latter. Meanwhile, fossil methane is of course a huge problem.)
As for the second part of your question: When it comes to meat production, we have already reached the steady state.
As for the first part: Yes. But it is not that high. Let's say the total amount of greenhouse gases in the atmosphere due to cattle (and rice and other types of agriculture) account for a fraction X of the world's yearly emissions. (In Germany, for instance, X = 0.06, i.e. 6%.) Let's assume for simplicity the remaining emissions are exclusively due to fossil fuels and all emissions are the same every year. Let Y be the fraction of agricultural emissions relative to fossil emissions, i.e. Y = X/(1-X). (= 6.4% for Germany)
Then, after N years, the amount of greenhouse gases in the atmosphere due to agriculture, relative to the amount of greenhouse gases in the atmosphere due to fossil fuels (burned during that timeframe) will be Y/N. This is because agricultural greenhouse gases in the atmosphere will stay constant – because we're in a steady state – but greenhouse gases due to fossil fuels will keep on accumulating.
Putting e.g. N=20 yields Y/N = 0.3% (for Germany). N=100 yields 0.06%.
In short, the amount is negligible. Fossil fuels are a real problem when it comes to climate change, agriculture is not.