Instead, use World, and cast to ICubicWorld when needed.

It wasn't really a good decision to do it that way. It provides little simplification while making it hard to use methods in World that aren't already there in ICubicWorld, and makes updates harder. Same should go for IColumn

Do you want it to be cast on every use or are stored casted references okay?

It really depends on what and where is used more frequently. If a class nearly exclusively uses ICubicWorld methods - storing ICubicWorld is ok. If a method uses ICubicWorld methods a lot - storing it i lical variable would be ok.