I am planning to buy this extension for making one of my Joomla website as central administrator. I will be connecting it to more than 15 website.

My question is can I choose to publish an article on any number of websites selectively, i.e. I may want to publish on 1, I may want to publish on 3, or I may want to publish on all 15. It could be any variation.

NO.

To do what you want, you have to use the "Article Sharing for JMS".
The article sharing works like PULL and not PUSH as you have described.
In other words, this mean this is the website that want to publish an article that is responsible to select from which website he want to read the article.
This is not the central point that decide where to PUSH the article but the slave site that decide from which site (PULL) the article.

You can moreless simulate what you want if you create some specific folder on your central point where you store the article to "push" on the slave sites.
In the slave site, you can decide to read the category form the central point and display it as a blog.
So that each time that you write an article that you store in a specific category, it is displayed by the website where you which to display it.