What general observations about life or human nature does the author make or imply? Do the characters make any such observations?
Shakespeare, like most writers of complex, sophisticated literature raises questions about human existence more than he tries to pretend he has answers. He reveals ideas and those ideas are explored and elaborated on during the course of his plays: ideas such as the nature of good and evil, the purpose of existence, the nature of love, life after death, rational thought vs. immediate action.
At least as far his plays are concerned, observations about life or human nature would come from his characters, anyway, rather than from the author's point of view itself.
That said, there are numerous observations about life and human nature made by Shakespeare characters. I'll just point out a couple:
- Macbeth says directly that life is meaningless in his famous "Tomorrow" speech in Act 5.5.
- Hamlet says that if not for the threat of an unknown afterlife, existence would not be worth maintaining in his "To be or not to be" speech (3.1).
Of course, implied observations exist, also (I'm trying to at least touch on the different topics of your question), such as Hamlet's revelation that kings end up worm food the same as everybody else when he "tells" Claudius where Polonius's body is.
Unlike most writers during the Elizabethan era, Shakespeare refused to focus on politics and religion in his writing. He and his characters achieve a universality this way, as they are not mouthpieces for polemic diatribe. He honors the Greeks, Romans, Judeo-Christians, and the secular. In this way, he is a humanist ahead of his time. Shakespeare somehow found a way to synthesize and honor all belief systems without alienating any.
Also, Shakespeare focuses on the eternity of art and love. Although I would not classify him as a Romantic, he did exhibit a idealism when it comes to love at all costs. Many of his characters demand true love, and are willing to suffer greatly if they do not get it. Whether it is Romeo and Juliet or Desdemona, Shakespeare's characters refuse to settle for lukewarm affection; instead, they demand, and are prepared to die for, passionate love.
In short, Shakespeare has the biggest heart, I think, because he worked in a collaborative industry--the theater. He surrounded himself with open-minded people of all classes. He did not publish his work during his lifetime; in this way, he guarded himself against artistic pitfalls of self-righteousness, hubris, and petty grievances.