What are these Soyuz Astronauts/Cosmonauts holding in their left hands before launch? I suspect it must be the pump for the active cooling system by pumping water similar to those in EVA suits, but it would be great if you could clarify that. But if that is the case, why don't they have them after landing? Will they leave them inside the Soyuz capsule itself after landing? During the launch, where will they keep these in the small crew compartment?
They're portable cooling units for the spacesuits, as you guess, but they aren't used aboard the spacecraft; instead, the suits are plugged into the spacecraft's environmental systems. They're only used between suiting up and arriving at the launch site, hence they aren't stowed aboard the Soyuz and aren't with the crew at landing time.
В руках у каждого из них небольшие чемоданчики - там находится система жизнеобеспечения скафандров.
In the hands of each of them are small suitcases - there is a life support system for spacesuits. What cosmonauts do just before the flight