How do glaciers move? Could Antarctica be separated?
Glaciology is a large field of research, and I cannot include all aspects in a short answer. I'll try to give some concepts that can help you in your understanding of the cryosphere. 

Glaciers are, per definition, moving due to gravity towards lower altitudes. There are some exceptions, e.g. rock glaciers are the stagnant remain of a moving glacier. 

Large bodies of ice are called either ice caps (if smaller than 50,000 km$^2$) or ice sheets (if larger). Today, we only have the Antarctic ice sheet and Greenland, but during times of larger glaciation, large parts of the Northern Hemisphere were covered. Within the ice sheets, fast flowing areas are referred to as glaciers. Where those areas are formed depends on many parameters such as the mass balance, the basal melting, the topography, etc. 

If the glacier or ice sheet reaches the ocean, it becomes buoyant. For larger glaciers, the section near the coast remains bulky enough to remain in contact with the ground.  The point of extent where the ocean reaches under the ice is called the grounding line - a subglacial beach, if you like. 
As it's floating, tidal variations moving the ice sheets help it to break up, but in some cases, depending on thickness and topography, it can float in a rather uniform shape which we call an ice-shelf. 

The same way as rivers and lakes are formed as rainwater floats towards the oceans, ice sheets and ice caps are formed as snow falls in cold areas, where it doesn't get warm enough to melt during the summer. As the snow accumulates it forms firn and later ice. The ice starts moving as the gravity and shear forces make it unstable. 

Eventually the ice will melt, either on the surface due to warm air, e.g. during summer months, or in the oceans as it calves off into icebergs and is warmed by the ocean. Glaciers can also melt on the base due to heat flux from the Earth beneath or due to pressure. In this way, subglacial lakes are formed under the ice. 

Large parts of Antarctica are situated below sea level. Partly this is due to the intensive erosion of fast moving ice, but the main reason is that the weight of the ice pushes down the crust into the mantle. If the ice sheet melts, the mass decreases and the land under it will rebound. This happens almost on a human timeline. Some areas, such as Hudson Bay, Scandinavia, and the Antarctic Peninsula, have a fast uplift making sea into land. 


Maps from Quantarctica. 1: Ice flow velocity. 2: The topography under the ice. 3: Geoid height (not so relevant for your question). 4: Snow accumulation and blue (=old, hard) ice.

As I've suggested before, I recommend you to download the Quantarctica package. It's rather big but free and easy to use. You can investigate how the ice flows are forming the ice sheet and how the bed map looks under. 

The large chunk of Larsen C Ice Shelf that just broke off, doesn't directly reshape the Antarctic continent or cause any change in global sea level, as it was already floating. However, if the ice shelf becomes more unstable, it can affect the ice sheet left trailing it and the flow pattern of the glaciers, and that would have a large impact on most human activities - and beyond.