How to handle falls in flooring ?

Yes, this is possible, but not necessarily simple.

One way to do this would be family tables, but the table would quickly get very large and burdensome. I think using a master part plus inheritance features in child parts would do the trick.  I'm not going to go into the how to on each feature I'm suggesting, I'll leave learning them up to you.

Start by creating a master part that represents the full board, out of the box.  Place that in your assy every time you need a full board.  

At the end of each row, create a new part and use the inheritance feature to import the full board part.  This will be a generic, name it something like row_1-2, meaning end part for row 1, start part for row 2.

In that part create a datum plane offset from the end representing the length needed for the end of the row.  Use the solidify feature to cut off the "fall".

Suppress that solidify and create a new solidify feature cutting the other way, creating the "fall" part.

Now, create a family table and add both solidify features.  Create two instances, one with the cut creating the end piece and one with the cut creating the fall piece.  You now have the parts you need for row 1.

For the remaining rows, you can simply make a copy of the initial master part, change the offset of the datum plane and rename the instances.  This will give you a cut piece and the "fall" for each row.  Depending on you accurate you are after, you may want to add a second cut representing the saw kerf as well.

I'm sure someone else might have a more clever way of doing this, but this would be my approach.  Interesting challenge.

Doug knows more about inheritence features than I do, so his answer will work.

I would look at the projetct and determine if this is a one-time thing or something that needs to be done on multiple projects. For a one-time project, I would just use family tables to get the different lengths and place them together to make the floor.

One other way to do it would be to just place full boards, and then trim them all to the room size. The fall of row A to be the beginning of row B would just be calculated and then that new board placed with an offset for the fall length.

You did mention pattern, if after placing X boards, you are back to starting with a full board on row Y, you could group all of the prior rows and then pattern them to fall into place after the Y-1 row.