No, we don't know all the variables and how they interact.

However, the models become more sophisticated all the time, and as they do they begin to converge on what are the most important variables and their interaction. And, the models are testable.

We can (and do) test them in several ways. The most obvious way is to simply use the model to project forward, then wait and see how well actual events agree with the model predictions. This works, and we do this, but there are some obvious disadvantages with this approach. For one thing, it takes awhile. Also, in the case of climate models, if we wait to see if the prediction was correct, it is then far to late to do anything about it.

So the other way climate models are tested is by "hindcasting". We select some point in the past when we have a reasonably good idea of what conditions were. Then we run the model forward to today. Then we look at how good (or not) the predicted conditions agree with what we can actually observe.

Our understanding of the variables and how they interact gets better, which allows us to better quantify them in the model. New approaches and techniques give us a much better picture of past climates, which helps us test and calibrate the models by hindcasting. We get more accurate and more complete data on current climate all the time, which provides better input into the models. And improvements in computer power enable us to run much more detailed models with finer resolution.

No, the models aren't perfect. But they get better and better all the time. And what they are telling us is not good.