How to wire Dining room to code: in an open concept kitchen, with one outlet, and stairs
Since this is a remodel of both rooms, it will require both rooms (or at least the scope of the remodel) come up to current code.  

The kitchen, dining and living rooms can share a lighting circuit.  

The two kitchen countertop receptacle circuits can't serve anything else.  Your cook will thank you if you use more than 2 circuits (to be more precise, you'll never hear about kitchen countertop nuisance trips).  Since you are adding these circuits, every point along the backside of the counter must be within 2' of a countertop receptacle. 

Similarly, every point along a dining room wall and a kitchen non-work-area wall must be within 6' of a wall receptacle -- in whatever area your AHJ declares is part of the renovation.  AHJ=permit issuing authority/Code inspector. 

A person at the bottom of the steps or the top of the step must have a way to turn on the stairway safety lighting from each end.  This is only an Code issue if the lighting has a switch.  In the LED age when a 40W equivalent light bulb costs $6/year to run and will run 25 years, having safety lighting on 24x7 or day/night sensor activated makes some sense, and then you don't need switches at all.  Code is not particular as to what stairway safety lighting looks like, but your inspector might be.