I think it could be correct or incorrect to include the definite article, depending on the context. As Soglio says, it would be correct to use the definite article when referring to a specific instance of joy. If we that not all joy is fleeting, then this example seems OK to me.
The closest correct sentence to what you wrote is probably Deshalb habe ich ein Problem mit dem Unterscheiden dieser Wörter (using the genitive case dieser Wörter to connect the object of the verb to its gerund [noun] form das Unterscheiden).
But in German, it's generally considered more elegant to use verbs as verbs rather than as nouns, so a better alternative would be Deshalb habe ich ein Problem damit, diese Wörter zu unterscheiden. or perhaps even better, Deshalb fällt es mir schwer, diese Wörter zu unterscheiden.
Not necessarily. If the reference is to happiness or joy generally, then yes, you would not use the article; you'd say "Happiness is brief." But if you're referring to the particular joy of the particular situation--e.g., "I was happy because I got a toy in my cereal, but the happiness was short: the toy broke."
Excellent explanation and rule of thumb. Looking at Philster's example below: Referring to the specific state of his family one is using 'the' : However, make it a general statement: "Happiness is important to my family" and you see that 'The' has been lost to make it grammatically correct.