Beautiful flooring! The dining table should have a square shaped rug.

Yes, beautiful floor! However, I would suggest a round rug under your round table setting. Make sure that it is large enough so that when the chairs are pulled out to sit down, the chairs remain on the rug, This takes at least 1 foot beyond the pushed in chair. So, if the measurement across the group is 8', you would need a 10' diameter rug. If you prefer a
square shape, the same holds true and you are then looking for a 10' x 10'.

As for pattern /palette, the rugs throughout your home should coordinate color-wise and style-wise. Or you could think about a patterned rug with a solid one in another space. Since you have kids, it would be nice to add a runner to your hallway for noise control! So, of the three rugs you are thinking about, 2 could match with another in a solid, or the third one could be the same pattern but with the same colors moved around.

It would help if you could post a pix of the rug you have in your LR so we can suggest options for the DR and hallway, etc.