How to download files which are not dmg files to Mac applications? Downloading files on Mac via .dmg is super easy, but anything else e.g. tar.gz files or zip files is completely baffling to me. Where do I unzip the files to make sure the applications will work and be added to my $PATH and $CLASSPATH? What is the correct process and best practice to follow for all of these formats?

A great example is me downloading this:

http://stanfordnlp.github.io/CoreNLP/

Where I don't know what to do next. Obviously I don't want to unzip into my Downloads folder? Do I unzip into my applications folder?
There is no general answer to this, it depends entirely on what is in the download.

In the particular case of the Stanford NLP download that you mention the zip contains resources for a Java developer to use when building a natural language processing program. It does not appear to contain anything that you can run without doing some Java coding.