FWIW, in the U.S., the rules of the Professional Skaters Association (PSA), which govern coaches, require that the prior coach of a skater be fully paid up before a new coach can take that skater on as a student. I'm not sure that Rudkovskaya's request is much different than this rule.
This has been discussed for some time - there is no contact system, similar to American, in Russia. Not in figure skating, at least. The discussion about necessity of contracts was fueled by Alyona's and Sasha's exit from Khrustal'ny. I'm sure the Alyona's case will create a precedent and things will change. I feel sorry for her right now. I'm sure Yana will find a way to make them pay - through court or in some other way. It's mesmerizing how them both, she and Plushy, say that that they parted ways with Alyona on good terms. And then - here we go, her mother has contractual obligations and a debt. They are not wrong - they are a commercial club that provided services to Alyona. I wonder if results is a factor in the contract. Probably not. Otherwise, Yana and Plushy would need to pay to Alyona and her mother.