Which would be the beginning of world government. Agreed upon rules with clear consequences for breaking them. Right now there are consequences for doing what the major players do not like, but the rules are not written down beforehand.
No it wouldn't, because it isn't a government, it's a relationship between fully independent states.
If a federal judge orders an Alabama school superintendent to do something as a matter of federal law, that superintendent is either going to do that thing or be arrested by the feds, and nobody is going to stop them. That's what a government looks like.
If the EU tells the US to cut carbon emissions or face sanctions and Donald Trump tells them to piss off, nobody is going to arrest Donald Trump and anybody who tries to is going to be met with force. Europe can then impose the sanctions, and the US can either give in or suffer the ramifications as a country, but at no point is anyone who refuses to comply going to be held personally responsible by an external governing body.
Moreover, each government's tools against the others are roughly symmetrical. If the EU wants to put sanctions on the US they can, but the US can just respond in kind. That hurts both of them and they both know it, so (absent corruption) they won't do it unless it's something sufficiently important. Which means you don't get the insane policies like we have in the US where the feds start micromanaging everything just because they can get away with it.
The problem with world government is that you would need sign on by literally everyone, permanently. As soon as you had a serious disagreement and tried to enforce it against a sufficiently powerful country without their consent, we would have at best the dissolution of the world government and at worst World War III. The American Civil War at world-scale is not a pretty picture.
The system we have now works much better than that. When someone is cheating, the countries that don't like it put sanctions on them until they stop. The sanctions aren't severe enough to fight a war over but they cost more than the cost of doing the right thing, which resolves things for all but the most obstinate countries.
The problem with climate change seems to be that countries lack the will to actually use those methods. If most of Europe and China got together and signed an agreement to all put sanctions on any country that didn't meet a per-capita carbon emissions target, that would be the end of it. But apparently they don't have the stones to actually do it.