The first and most important reason is that standard regression models had a one to two-hundred year headstart on copula models (depending on exactly where you count the genesis of regression models and copula models). Any explanation is the disparity in usage is going to have to start there.
The method of least-squares estimation for fitting functions through data was developed in the early nineteenth century by Legendre and Gauss, and the Gauss-Markov theorem was published by Gauss in 1821. By the late nineteenth century the term "regression" had come into use to describe the narrow phenomenon of regression to the mean, but it was developed further at the end of the nineteenth century in a form that is a clear precursor to the modern theory. In particular, Yule gave a close precursor to the modern regression model in Yule (1897) and Fisher had developed and analysed the standard Gaussian regression model that is used today no later than Fisher (1922).
Contrarily, copulas were first introduced into statistics in Sklar (1959) and were developed further over later decades. The initial mathematical result underpinning the field was a "folk theorem" for over a decade, until it was proved by multiple authors in the 1970s. The first statistical conference looking at copulas didn't occur until 1990 and even after this, copulas were only really applied in the field of finance. Copula models did not really become widely visible in the statistics profession until about the turn of the twenty-first century, when Li (2000) popularised them in a seminal article in finance. It is probably only in the last two to three decades that copulas have become broadly known even within the statistical profession. As you point out, the copula theory is mathematically more complex, but it is also much, much younger.
Statistical theories and models tend to start out with narrow usage confined to scholars in the field and then --- if they have sufficient value--- they expand out to be used more widely by various professionals in a broader range of applied fields. It is not until they become sufficiently widely used in the professions that universities decide it is worth teaching those models in their regular courses. In the present case, copula models are about twenty years old and they have probably only started being taught in the universities in the last ten years (and at some universities not yet at all). You only have to go back about a decade and statistical students at a university would not even have heard of copula models (unless they ran into them as a speciality) and would not have had any courses that taught it.
So, if you are a statistician/econometrician and you are over forty, you probably will not have learned about copula models unless you have personally gone out of your way to self-learn it outside of your university education. However, you will have had at least a few courses that covered regression modelling, GLMs, etc., and you will have had to implement these models regularly as a student in order to complete your degree. If you are a psychologist or scientist over forty, you almost certainly never learned copula models, but you probably would have encountered regression models in your university training. This has a huge impact on the respective level of usage of the two models in subsequent professional work.
Best Answer
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:
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.