Dividing a table between two pages.

Does this split columns, or [just] rows?

Also, if it splits columns too, do they lose the flow as one table?

It is not possible to separate the table horizontally. But, If this is important for you to do in that way then try to rotate the frame vertically and youi will get the Red Plus sign in that way you can split it... But this is not the right way of working.

Create TWO separate table or Instead of inserting columns try to insert more rows and separate it to the next frame.

tx

In such cases I draw a text frame accross both pages of the spread. I insert in the spine area a empty column with no color. 
It is faking 2 frames, but it works. 

You can’t. Tables can only be split between frames verically. The position of the frames is not the issue (that is, you can place the frames next to each other) but the break between text in frame one and text in frame 2 is vertical. If the yellow cells were separate rows instead of separate columns you could do it.

A workaround would be to insert a column between the blue and yellow columns and put the table in one frame. The inserted column would act as a gutter between the two parts of the table.