The B sample is tested when the athlete in question requests the test - I don't know how often it happens that an athlete would just accept the results of the A sample & admit to doping. We don't know how long it will take, but given the very high profile of this case I would think that test is being done as quickly as is technologically possible.
RUSADA suspended Valieva when they got the test results, someone on her behalf (because she's 15) appealed, and RUSADA lifted the suspension. WADA, and the ISU are part of the appeal, but the IOC is the primary appellant, represented by the ITA, because they want the suspension reinstated because the Olympics are currently ongoing, and yes, they don't want her to compete and then have to change results later. The IOC has taken a lot of steps over the past several Olympic cycles to speed up the process of doping suspensions during the Games, when people are paying attention, to ensure the proper athletes get the medals they deserve rather than receiving them some time later.
I hope all of that is clear, and if not I will try to clarify.