My understanding is (I read it somehwre here or on the other forum) that after worlds Elena and Nikita were supposed to go to train to Zoueva, so maybe the Federation decided that the new pair (not officially confirmed pair) should go there to see how it goes. I can't see why it shouldn't be legal that Victoria and Nikita went to Zoueva. If I wanted to go to Zoueva (and had a partner) and she took me, that's a contract between me and Zoueva. If Russian Federation decides not approve them, they can refuse to fund them, but there is nothing illegal on a person skating with another person.
I don't know who pays for it, but I am sure the families can afford to pay for their children to train somewhere for 3 weeks. After all, most families pay for their coaches all year round. It is only in several countries that skaters get some funding. Most countries can't afford it and the skaters/their parents pay themselves (sponsors, or parents doing several jobs, or skaters having some part time job e.g. teaching skating classes, or remortgaging house if they have to). Besides, Nikita's family may not be rich, but Nikita for example brought a gold and bronze medal from Sochi and got a car as a reward. If they were really struggling financially, I am sure he could sell the car! But there may not be need for that; he must have some sponsors. Skating is popular in Russia and he brought two medals from Olympics.
If it was between kids and Zueva, then it would be OK, but it wouldn't be logic while they go there for three weeks only - like why they would ask her for help for just three weeks - so it looks like Russian Federation's decision.
My question was whether it would be legal for Russian Federation to send and finance a tryout of couple who is no more in National Team and who officialy does not exist. If Mutko asked Federation later to explain how was certain money invested, what they would answer him?: "You know, it was for two people who were not a couple and not in National Team, we only wanted to know if they can be compatible and we were not able to find it ourselves in few days (because we have only Tarasova, Zhulin, Svinin, Zhuk, Bestemianova, Gorshkov, Tchaikovskaya, Kustarova specialists), so we send them to Zueva." Like Russian Figure Skating Federation would finance a trip for Ovechkin and Malkin who wanted to go to Zueva for some skating classes...
I never heard that Nikita would have some sponsors, I know Lena has...she noticed it somewhere (she also noticed that it was very hard to get a sponsor). I don't know if she meant Pantene and P&G or somebody else. But if it would be those two companies only, then it would be Lena's sponsors, not Nikita's.