Perhaps the best way to do this is to create a Node class with name, gender (+other) attributes, and also a VerletObject attribute?

To answer your title question: no.

You can't be a Dog and a Cat at the same time, even though both extend Animal. You can be an Animal and a Canine and a Dog, because then you are in a hierarchical OOP inheritance chain. Both Dogs and Cats can have a Tail. But then it would not make sense if Tail extends Animal. From your initial description it sounds like either you are trying to create DogCats or TailAnimals.

If "a node is not a physical object" then why is it extending PhysObj? One could opt for an extended version of the VerletObject (with additional attributes such as name etc. or just a Node attribute) but then of course you should extend VerletObject (this is the most practical route if you are using Toxiclibs VerletParticles). Looking at the whole however, your last suggestion sounds the most logical to me... a Node with many attributes among which VerletObject is one. But as said, this choice depends on conceptual, practical and personal considerations.

Good answer from amnon.owned (as often!). It is called multiple inheritance, and even if there can be a logical reason to do this, Java just doesn't allow it.
Note that inheritance isn't the only way to do OOP (and some are even reluctant to use it). You can also use composition, ie. instead of making a is-a relation (dog is an animal), you can make a has-a relation (dog has a tail).
For this, you just need your class to have a field of the type having the properties you need to use.

Thank you guys! I'm getting the feel of OOP now!