Climate change research focuses on hindcasts and forecasts with models, but also extensively with observations and theoretical understanding of the present climate system. So, yes, we use statistics in our predictions (of future and past), but also regularly analyze the physical climate system. I don't think it's fair to say studying the climate is purely statistics. I think you could perhaps make an argument for specifically 'climate change (prediction) research.'
This is simply untrue. El Nino forecasting, for instance, is based on statistical models. However, global climate models and regional climate models use coupled dynamical cores that are completely physics-resolving. We don't do statistical modeling. We literally run a dynamical model with full physics forward in time and look at what comes out.
The 'experiment' part is tricky. There are experiments to prove the greenhouse effect (by adding CO2 to a flask and maintaining radiation). There are in situ experiments (perhaps better said as 'observations') of the climate system where we have Lagrangian drifters move through the ocean to take samples, or have Eulerian instruments set up for our global weather observation network. There are A/B experiments where we can put mesocosms into the ocean and perturb things and test hypotheses. In my mind, climate models are great for experimenting. They hold a dynamical core of primitive equations, and we can run control runs, and then crank up CO2, or add iron to the ocean, or change parameterizations, etc. and see what happens. The observed warming in the real world is only reproduced by models by running an 'experiment' with enhanced CO2 in the atmosphere.
I don't see what you're getting at here. The climate system, and weather, are characterized by chaos and inherent unpredictability (a la Lorenz). We make probabilistic predictions of the weather, knowing that some will fail. Rain isn't a great example either, because there is considerable debate in weather forecasting of what X% chance of rain means (over what area? over what time span? etc.)
Improving the dynamical cores of weather models has produced one of the greatest examples of increased forecasting in the modern world (see this figure:
https://goo.gl/x7i2k1). Hurricane forecasting as well. This is in stark contrast to earthquake forecasting. Nate Silver has a great chapter on this in his book.
I pointed to experiments earlier in the sense of climate models unable to reproduce warming without CO2 forcing. This Bloomberg article also does a great job:
https://www.bloomberg.com/graphics/2015-whats-warming-the-world/. But that is based upon residuals and statistics, which you might find unsatisfying.
Evidence-wise, anthropogenic carbon has a distinct delta13 signature we can measure in the atmosphere that links the rapid rise in CO2 to human activities. We can also track this unique carbon (as well as 'bomb carbon') as tracers in the ocean as it trickles into the deep via "deep water" formation sites.
Further, atmospheric oxygen has declined proportionally to our rate of fossil fuel burning, which would only happen upon combustion. If CO2 were rising naturally, you wouldn't see a subsequent decrease in O2.
I agree with you completely here! I'm a big fan of getting rid of the consensus argument.