In wiki (https://en.wikipedia.org/wiki/Covariant_derivative#Tensor_fields), it mentions that the covariant derivative of tensors satisfies
$$D_v(R\otimes S)=D_vR\otimes S+R\otimes D_vS$$ and $$D_v(R+S)=D_vR+D_vS.$$
How to derive $$D_v(R(X_1,X_2,...,X_n))=(D_vR)(X_1,..,X_n)+R(D_vX_1,X_2,...,X_n)+R(X_1,X_2,...,D_vX_n)$$?
This does not follow from the two properties you list (and only the first of this two properties is actually needed in what you want to show). You in addition need that the covariant derivative commutes with contractions. Knowing that, you just have to observe that $R(X_1,\dots,X_n)$ is the complete contraction of $R\otimes X_1\otimes\dots\otimes X_n$ and use linearity of the complete contraction.