I have no idea- sorry! But I just wanted to let you know that there is a thread on this board called "drape gate" about a small dining space. It's epic. Thousands of posts. I have learned a ton from that thread and it may give you some ideas on smaller (not that 13x13 is small...mine is much smaller...but room is thread is around. 11x11). Good luck

It depends on what else is in the room and where the openings are. Ideally, each chair would have at least 3' behind it when pushed all the way in.

You can go to an Icovia room planner and play with different sized dining tables and sideboards and chairs here: http://roomandboard.icovia.com/default.aspx?projGuid=B4DB5B9A-5C61-4E9E-A2DB-DBCB79B72CAB

8 foot diameter table would provide plenty of room when entertaining.

It does depend on what else would go in the room...most times there is some kind off buffet which will close in the area where the dining table and chairs would be placed and you will need approximately 3' around the area for using the table so...maybe a slightly smaller table like 7' would be best...one way to really see the area is to do a template and see what you think or have it drawn onto CAD.