Is it possible to make the KA version of Processing.JS library available so that students would be able to run their programs offline without much trouble?

Yep! You have two primary options, though neither is convenient or easy. The first one is to download KA's entire live editor, in order to provide as close to KA's editing experience as possible. This option gives you code on the left, output on the right, and error pop-ups, same as you get on Khan Academy. The other option, if you just want to use Processing.js, is to create an HTML file, load in Khan Academy's version of Processing.js, and then include your code. Then, opening the file will give you the output of the program in a browser window. The second option is more flexible, but also more technical and farther removed from the experience you'd get on Khan Academy. The former option is simpler, but you lose flexibility (for example, changing the canvas size), and it's harder to create and edit multiple programs. If you want the best and full Khan Academy experience, you'll still want to continue using Khan Academy online. Neither of these options is official support, and both would be considered hacks or workarounds. If you're adventurous, proceed.

Khan's live editor is open source on Github, so you can download the code that powers it from here. In the upper right there a button that says clone or download, you want to click that, then hit Download ZIP. Open the .zip file in your default downloads location, unzipping it as necessary. Then navigate inside of that folder to /demos/simple and double click on index.html. This will open the version of KA's editor that you just downloaded in your default browser and you'll be free to write code as if you were writing it on Khan Academy. You might notice, there's no save button, however, because normally programs are saved online on Khan Academy. This mini-editor can save one program in the caches of your browser, so you can close it and come back, but if you want to store more programs offline, you'll have to create files to store them in. For example, create a .js file with Notepad (on Windows) or TextEdit (on Mac), copy and paste your code into it, and save it. Then open the file with the aforementioned text editor and copy the code out of it into the editor again. Far from ideal.

To create a program that can run more independently, you'll need to port your processing.js file to HTML. Normally there are a couple of steps in downloading processing.js and creating a new file, but I've created an example folder for you to download. From here, click download ZIP in the upper right. Then you need a text editor that can write code without messing up the formatting. Notepad (on Windows) or TextEdit (on Mac) are some pre-installed options, though you'll need to to follow these instructions to use TextEdit. (If find yourself using either of these often, you should look into downloading a better program, like Atom, Sublime Text, Visual Studio Code, TextWrangler, or Notepad++.) Using your text editor, open up "example.html" inside the ZIP you just downloaded, unzipping it if necessary. You'll see some Processing.js code there, under the comment "// Program Code Goes Here". To see the output of this code, you just need to open this html file with a web browser, and you should see a big face smiling at you. To create more programs, you can duplicate and edit this .html file, as long as they remain in the same folder as processing.js, opening them will run your processing code correctly. 

Unfortunately, this is not an easy process, and it's not one that KA really wants you to do. If you run into difficultly, I will be available to help answer questions. Hope this was somewhat helpful, and I'm sorry for its complexity,
-Matthias