That sounds like a fab idea, the only thing I don´t know is whether you would need to add more sugar to prevent further fermentation.....someone clever will be along in a minute!
I would just boil the jar for 10 minutes.
To stone or not is your preference. I do, but that´s because I want to use them in cakes and can´t get my head round the idea of eating cake and spitting out cherry stones, if you are going to use them for cocktail purposes 

  then leaving the stone in would be more acceptable.
The recipe I followed for mine was based on the authors grandmas cherries some of which remain and are over 40 years old. Apparently by that time even the stone is infused with the alcohol.
baby-loving, earth-digging, bread-baking, jam-making, off-grid, off-road 21st century domestic goddess....
...and 
eco campsite owner