I don't think so. I've never used that sentence without the definite article and I've never heard someone else missing it. If you don't want to use the article 'la' you can use a variant like 'preposizioni articolate', for example: 'Lui ha dell'acqua nel bicchiere' The same applies to 'Avete cartoline?' -- 'Avete delle cartoline?'
I don't know. Maybe sometimes they use articles to define it more clearly, like "l'acqua" means "the water" but "acqua" means "water". Although when the phrase is "l'acqua" you can also translate it into "water". I'm not sure. Italian is certainly very confusing, as all language goes!!
Because the app wants you to specify the presence of the water "in" the glass. It can roughly be translated as "He has a glass of water." in some other context, but here you should translate exactly as it says. Plus, if you need to say "He has a glass of water.", you can say "Lui ha un bicchiere d'acqua." :)
This is strange for speaking exercise. Even if you don't speak full sentence it becomes correct. Once I purposely coughed instead of speaking the sentence.. and still it was correct. So speaking exercise has no proper evaluation in duolingo
I can help you for the first question. Nel and nello are both abbreviations for 'in the'. The difference depends on the version of 'the' used for the word. If the word uses il like bicchiere then "in il" is shortened to 'nel', but if it uses lo such as in "in lo zucchero" then it's 'nello'.