The least arbitrary species definition says that members of the same species are capable of producing fertile offspring of both sexes, i.e. we're talking one gene pool. As soon as they can't interbreed and produce fertile offspring, we're talking about different gene pools. A speciation event is when two separate gene pools arise from one; "separate" meaning that genes aren't interchanged between the two pools anymore. If speciation couldn't occur, evolution would be impossible. Before speciation had been observed one could make reasonable arguments against such a possibility, because the continual mixing of genes within a gene pool would slow divergence down, and one could propose selective pressures against speciation events, because animals that has diverged too much from the rest of the gene pool wouldn't be able to produce fertile offspring with anyone. Speciation has been observed though, and each new species will now be able to evolve down their own path, i.e. macroevolution.
There are indeed disagreements, and that's what makes science work. I find it curious however that your examples use varieties and species interchangeably. In my understanding, when scientists use the word "variety" they're generally not talking about species, so the terms are not interchangeable. So it's not a contradiction to say that there exist one species and 10 varieties of red fox.
We've changed them so radically from their wild ancestor that they look nothing alike it anymore; no wonder these types of debates happen. Funny how radically life forms can be changed, isn't it?
It's a fun animal for sure. But it doesn't have "the fur of an otter, the tail of a beaver, the bill and feet of a duck, and the venomous spurs of a fighting gamecock". For instance the bill only looks superficially similar to a duck's beak, but these features are very different when you study them closer.
Peter