For questions like this, I normally agree with Barrie's comment, but I'm having trouble figuring out why a test would ask you to differentiate between these largely synonymous words with such examples.
The words have somewhat circular definitions in dictionaries. For example:
propagate: to promulgate; disseminate [Collins #3]
disseminate: spread or disperse (something, esp. information) widely [Oxford]
spread: to give information to many people [Macmillan #7]
Disseminate means to spread, but the implication is to spread widely among a large group of people. Still, in the example sentence, I think the word broadcast would fit better than disseminate.
Propagate also means to spread (Macmillan describes it as to spread ideas, beliefs etc to a lot of people), but it also carries a connotation of passing along information. You hear something, and, instead of keeping it to yourself, you tell someone else.
Spread is the most generic of the three, and the word can be used in a wide variety of other contexts (you can spread peanut butter on a slice of bread, for example).
Idiomatically, spread is often used with rumors, but that doesn't mean the other two words are never used. Here is the Ngram for “propagate rumors” vs. “disseminate rumors”; the expressions are found, but are dwarfed when we add “spread rumors” to the Ngram. That's the main justification I can think of for using spreading with the third sentence; rumors are generally said to be spread.
All in all, though, I don't think the answers to sentence examples are utterly obvious, even to a native speaker, and you can find contextual examples of the words being swapped. For example, reports can be disseminated, and news can be propagated.