Minimum height of step between two interior rooms

This can be a tough problem to work out in a situation like this. Particularly if you are trying to run the same type and size of molding from one floor area through to the other. What I have done in instances where the height changes are small is to simply trim the bottom of the molding so it fit into place over a distance such that the it slopes from one height to the other. If the slope is small and over enough of a distance it will never be noticed by anyone but you (unless you point it out to others :^) ).

Installing the sloped molding piece does impose some interesting work with making the miter cuts on the ends. It may also involve some planing and sanding of the molding surfaces at the joint to blend it around the corner. (BTW you will be a long way ahead with this type of "blend to fit" with real wooden baseboard. MDF or plastic foam type baseboard fails miserably when this is attempted).

I see that the existing darker colored floor has baseboard without any base shoe molding. You may want to consider the addition of base shoe molding in the new area because it can play an essential role of covering up what you have to do along the bottom of the baseboard to slope it from one height to the other across the transition zone. It is acceptable to bring the base shoe molding up through the transition zone and then just cut it back off at an angle at the end of its run.

This answer depends on where you are in the world and if you are in the USA what state you live in.

Section R311.7.1 IRC 2009:

Stairways shall not be less than 36 inches (914 mm) in clear width at all points above the permitted handrail height and below the required headroom height. Handrails shall not project more than 4.5 inches (114 mm) on either side of the
stairway and the minimum clear width of the stairway at and below the handrail height, including treads and landings, shall not be less than 31.5 inches (787 mm) where a handrail is installed on one side and 27 (698 mm) where handrails are provided on both sides. Exception: The width of spiral stairways shall be in accordance with Section R311.5.8.

Above snippet from: http://www.co.stevens.wa.us/landservices/documents/STAIRWAYSECTION.pdf