Formally, "that money makes happiness" in the first one is a finite CLAUSE. Thinking of it in terms of WORD classes doesn't help. This syntactic constituent doesn't bear any resemblance to adjectives and hardly any with nouns (only insofar as it specifies the head noun, in a manner similar to noun phrase appositives)
"That money makes happiness" is a type of a complement clause that is very common not only with nouns but with verbs and adjectives as well. Your sentence can be rephrased as:
I don't believe that money makes happiness.
In this sentence the that-clause is made a verb complement, instead of a noun one in the original sentence.
The that-clause complement of a noun shouldn't be confused with a relative-clause noun modifier. The function of the that-clause noun dependent in "I don't agree with the idea that money makes happiness" is quite different from that in "I don't agree with the idea that she proposed".