Weighted estimates are obtained for the derivatives in the model (shiftcoinvariant) subspaces (formula presented), generated by meromorphic inner functions ⊝ of the Hardy class H^p(ℂ⁺). It is shown that the differentiation operator acts from (formula presented) to a space L^p(w), where the weight w depends on the function ⊝', the rate of growth of the argument of ⊝ along the real line. As an application of the weighted Bernstein-type inequalities, new Carleson-type theorems on embeddings of the subspaces (formula presented) in L^p(μ) are proved. Also, results on the compactness of such embeddings are obtained, and properties of measures μ for which the norms · L^p(μ) and · _p are equivalent on a given model subspace (formula presented), are established.