Placing stove in center of island or on an edge
It is a question I deal with all the time as when I tear down walls between a kitchen and dining room I often add an island.  The answer is really simple for this.  The stove goes in the middle.  You need to find 12 inch cabinets.   

There are two issues here.  First the sides of ovens look crappy.  Makers don't spend time on the sides so you don't want it showing.  Second is if you hide the side with a wall then how do you construct this, what sort of countertop goes on top of the wall... it is just odd.  

First I have run across almost your exact use case before but I generally would tend to not try to put a stove on an island that small.  But you can still do it and it can look really nice.  You will need to buy a slide in oven.  Run the countertop full length with the over cut out - so you should have counter behind oven.  Let the counter overhang on the sides a couple inches to provide prep space - 15" is a good amount of space.  

For your 12" cabinets get a drawer one cabinet combo and put in full width shelving that pulls out all the way.  The right pull-out shelving can make a 12" cabinet very functional.