It's possible for very short, careful moves, but it's risky. Vibration from a longer carry or truck ride can crack unsecured glass even if the doors stay latched, so removing loose panels or taping and blanket-wrapping fixed ones is the safer approach for anything beyond a same-room shuffle.
Most wood china cabinets weigh between 150 and 300 pounds empty, depending on size and whether they're solid wood or veneer over particleboard. That weight, combined with a high center of gravity, is why two movers and a proper dolly are recommended even for the frame alone.
Yes, whenever the cabinet allows it. Splitting a china cabinet into its top and bottom sections makes each piece lighter, narrower, and far less prone to tipping through doorways and stairwells than carrying it as one tall unit.
Remove every shelf, wrap glass or wood shelves individually in moving blankets or thick packing paper, and stand them on edge in a box sized to fit, never laid flat. Label the box clearly so it isn't stacked under heavier items.
Experienced movers typically will, if the cabinet has a removable hutch, doors, or shelves, since disassembly reduces the risk of cracked glass and makes the piece easier to maneuver through tight spaces. It's worth asking your moving company directly what their process is for glass-front furniture.