In my opinion the two methods (copula, regression) answer quite different questions. The copula approach is much more general than regression and one of the reasons why you have not seen regression models based on copulas, might be that using copulas is much harder than using regression. Two observations why this is so:
- For a copula fit you need to know or estimate the joint distribution of all variables involved. You do not need this for regression.
- If you are only interested in the response, regression gives you the answer more or less directly. But from the joint distribution you need to manufacture the conditional expectation of the response with additional effort.
This extra effort for estimating the joint distribution and only then finding the expected response would need to be justified by the specific problem you are interested in. Two justifications I can think of are: You are actually interested in the joint distribution (that is what you called "traditionally") or you know that your model does not allow for the standard assumptions of regression (additive independent errors, say).
On your questions 1. and 2.: Sure you can do this in theory (if the copula is differentiable and has a density). If you know the joint distribution, you can calculate all marginals and conditional expectations. The problems start when you want to estimate this from data. Unless your problem prescribes a specific, nice parametric copula, you might need special samples or lots of them to do this.