Haha.. I love the title...
1) The null here would be: " there is no difference between the methane emissions in the burps and flatulence of the cows in the control and intervention groups, i.e the difference between the outcome variable in the control and the intervention group is zero.
2) I agree with Colette and Margaret... a randomized trial would be the best design. I did read the abstract of the published article. The study actually used cattle from the same farm, of the same age and weight, and randomized them to receive the different diet options- this kinda makes sense because having cattle from the same farm would help with any confounders- and increase internal validity... If they are planning a larger more generalizable study, I agree with Margaret- a cluster randomized trial would be quite good.
3) The exposure here is diet mixed with seaweed. Upon reading the guardian article you would think that its a binary variable yes/no- 'yes' being the intervention group and 'no' being the control group. (However, in the actual study they do have 3 groups- control, a low seaweed and a high seaweed group- so basically 1 control and 2 intervention groups). It would then be classified as an ordinal variable- since there is an order to the categories.
The outcome has to be a quantitative variable- the article says the intervention cows belch out 82% less methane- which means they have quantified the amount of methane.
4) Now the thing is I read the original paper and know that the cows were in 3 groups. So, I would be thinking about doing the ANOVA (rather than t-test) since we need to compare means across 3 groups, or else a linear regression model if there is a concern for any confounding.
Anybody interested in reading the original paper, here is the link...
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0247820
Fathima