In fact, your long reference is the wrong way to do it, in several ways...
First, because you make direct use of fields, where the Java way would be to use getters. That's a minor point.
Second, because you hardwire the hierarchy of your classes. If you ever have to change this hierarchy, that will make lot of code to change.
Another minor point: the usage of this is quite pointless. Some Java programmers like to explicitely spread it over all their classes, but the majority just omit it, because it is implied and adding it adds noise (gets in the way of reading).

The context solution is close of the common solution used by many Processing coders. The most common solution is to pass a PApplet parameter to the classes' constructors and to store it in a field of the class. Thus, no deep dependency on the hierarchy, no local variable to create when needed, it is just available anywhere in the class.
Another possible solution, if you don't want to "pollute" the constructors is to create a class with a static field holding this PApplet reference. You init it at the start of the sketch, and you access it anywhere with a static getter, like SketchRef.getApplet(). This class can be a singleton: a class with private constructor to prevent instantiation.

Good point on hierarchy weakness. This also caused much fear in me. But the reason why I am not using a reference to PApplet is that I do not want to fill much memory with same values of the PApplet pointer since the number of objects may be immensive. Thus I would prefer to use the singleton method which also seems to be easily manageble and extensible for storing other global things in the future.

No problem for 'this', I won't discuss taste and style, just wanted to point out it was optional.
And don't fear for memory issue, it is just a reference after all, so duplicating it shouldn't consume so much memory. Well, you can even make it a static variable in each class, so you init it in all the various classes in the setup (thus no need for this parameter in the constructors), and get only one reference per class, not per instance. An interesting compromise between convenience and memory saving.