Edit: lots of articles about this online, but e.g. http://time.com/5575523/climate-change-inequality/ has some sobering statistics on the likely impact of climate change: e.g. "A 2015 study in the journal Nature projected that the average income in the poorest countries will decline 75% by 2100 compared to a world without warming".
Resource consumption happens by a person. There's only so many resources a person will use while being rich. A rich person might consume far more resources than a poor person, but rich people collectively have a much smaller impact than the poor, because there are far fewer of them. This means that if you start tackling climate change and the impact is that food becomes more expensive, then poor people will be far more affected by this than rich people.
https://theconversation.com/emissions-inequality-there-is-a-... has the Oxfam discussion.
There would be an impact on poor people as you say, but this depends on the exact policies adopted. A study such as the one summarised and linked here: http://bruegel.org/2018/11/distributional-effects-of-climate... would be needed to confirm either way - their conclusion is that climate change mitigation efforts are potentially but not necessarily regressive with respect to wealth distribution.
Easy. This has actually been done many times during Cold War.
You do it yourself, and so well, that their citizens will demand it too.
There were many social advances in the West that were imported (usually in a crippled way, but still) to the communist countries (I was born in one). Things like 5-day work week, education improvements, recycling, ecological laws, nuclear proliferation treaties... usually it's very small things, but they do make difference.
What you suggest only works if you're talking about improving quality of life.
> has their people eat as much meat as they want
It's not healthy to eat that much meat anyway.
> then the US would be powerless
And you shouldn't panic. The US is far from powerless. There are many countries in Europe that for example do not have nuclear weapons. Does it make them dead? No, they continue to live, in fact often with high quality of life.
This worry that you somehow "lose the race", it's such a nonsense (reminds me of "mineshaft gaps" from Dr. Strangelove).