Yes.
Yes.
No.
Not really.
Because...
Now to be more verbose...
Basically, Processing can be seen as a Java library: some people just write Processing programs in Eclipse and are happy this way.
But Processing is a bit more than that: it offers a PDE, a kind of simplified IDE, that compile sketches (Processing programs) and run them, and offers some extra tools like generating bitmap fonts (.vlw files) and exporting a sketch to a .jar or a .exe (or .app perhaps, and similar depending on platform).
The latter (export capability) is probably one of the reason why there is a download per platform. And perhaps some other dependent code: in libraries\serial\library, I see a rxtxSerial.dll (but also a librxtxSerial.so, strangely, and lib\export has a skeleton.app which probably targets MacOS. So separation of system-specific binaries doesn't seem applied strictly...).
So, to answer the third question, you cannot "just download a jar" but you can download the full distribution and pick from there only the jars you need (core.jar at least, some libraries, etc.).
It is not "another language which runs on top of JVM", it is almost pure Java...
The devil is in the details. "Almost" means the sketches' source is pre-processed (parsed and slightly modified), to make a full .java file from the .pde files (adding a
class <sketch name> extends PApplet around the code, a main() method, managing imports, adding F (float modifier) after all literal numbers with decimal points, changing 'color' keyword to 'int', and some other changes that aim to simplify writing sketches: after all, you can try a couple of lines in the PDE and run it without even saving them! Behind the scene, the PDE saves the file, process it, compile it and run it while capturing the outputs.
Not a real new language (like Groovy, Scala, Clojure, Fan, JRuby or JPython would be) but something slightly different from Java.
I hope I clarified some concepts (and that I didn't make mistakes...).