From a lecture series at Georgia State University:
The term simple sentence can be confusing because of definitions that many of us have learned that focused on the meaning or content of a sentence. Those misleading definitions said something like "a simple sentence has only 1 main idea." But that definition is just impossible to apply because it's impossible to be sure what "1 idea" is. A simple sentence can be short and with uncomplicated ideas--but a simple sentence can be long and with complicated ideas.
It goes on to present some examples:
We have lost millions of manufacturing jobs to Mexico, South America, and Asia.
Moving up the class ladder also brings unexpected costs.
And then adds:
Thus, we need to use a definition like the one given in the Longman Grammar. A simple sentence is a unit that has only a single subject and a single predicate. The subject can be compound, but the simple sentence is a single unit as in the following examples.
Which means there can be simple sentences with compound subjects:
Jack and Jill went up a hill.
Sociology and anthropology are both social sciences.
If a sentence has more than one clause, then it is no longer a simple sentence. And a compound independent clause contains coordinated independent clauses:
(He was crying) and so (I gave him back his jacket).
Here, I have added parentheses to show the independent clauses and the coordinating conjunction.
Returning to your example sentences, I can look at them in two ways:
He is good and happy.
He is rich but unhappy
Here, the conjunctions don't join independent clauses but "compound adjectives". This makes them simple sentences.
But they can also be viewed this way:
(He is good), and (he is happy).
(He is rich), but (he is unhappy).
These could be longer forms of your sentences, which would make your shorter versions elliptical constructions.
Is there an expression good and handsome that can be used (perhaps idiomatically) as a compound adjective for somebody? (The phrase good and ready comes to mind.) Or, if not, can it be treated as such here? It doesn't seem unreasonable, which would suggest it's not part of an elliptical independent clause.
Therefore, he is good and handsome at least seems to be a simple sentence.
In your second sentence, it's more unlikely that rich but unhappy could be treated as a single adjective. It's not the same kind of thing as the compound noun salad without tomatoes. Also, the sentence contains a comma after rich. This is more suggestive of the use of an elliptical independent clause.
Therefore, he is rich, but unhappy at least seems to be a compound independent clause (and, therefore, not a simple sentence).
What about He is tall, dark, and handsome? Can the three adjectives still be treated as a kind of compound adjective? Or must the sentence now be an elliptical compound independent clause? Part of me wants to say that it's still a matter of intended meaning and semantic function; however, another part disagrees. But, in the end, I'm not sure if everything is always black and white.
As an additional note, I found a site that gives the following as a simple sentence:
Mary and Samantha arrived at the bus station early but waited until noon for the bus.
Its analysis is that even though the sentence seemingly has two verbs, it's really a compound verb (no doubt because of the coordinating conjunction but):
"Mary and Samantha" = compound subject, "arrived" and "waited" = compound verb