The shading and color might be caused by SkyLight placed in the scene. In order for you to test if it is the root cause, you just disable it in the level and play again. In multiplayer (the built-in one like in the video) we just call the recapture method, so both server and client are updated same way, is important, because the SkyLight is the indirect light in the scene and if you have the directional light moving the indirect light must update accordingly.
I never used the split screen , so I will really need some time to look and understand whats going on. If it uses the built-in multiplayer in a special way, then there must be a way to fix everything. Actually it is hard to find someone with a good idea for split screen usage, so many assets probably are not made with that usage in mind Im afraid.
Anyway all is part of the learning process, but in my opinion you are going to stuff a bit complex right now… it is really important to understand how materials works in the engine, since it is the base for everything else.