More than one column. There is a load multiple column command. This loads columns that are contigous to each other. Usually the reconfiguration process loads one column on the left side of the die and then one on the right side of the die. This is to distribute the current evenly over the chip.
A full configuration loads one side then the other.
You answered your own question. The answer is "yes." The "static" bits of the design would just get reloaded as they are.