The sentence that requires translation is "Το παγωτό είναι γλυκό." It includes the direct article. Therefore, it's referencing a specific ice cream. The answer should also include the direct article. Because it's a word-select question/puzzle translation without the article as an option to select, it's incorrect.
Technically, dessert is "επιδόρπιο". Sweets can be eaten as dessert, but you wouldn't call a healthier dessert (for example fruits with yogurt) "γλυκό". So, they shouldn't be used interchangeably, even though "επιδόρπιο" isn't used as much, and - let's face it - we most probably are going to choose a sweet for dessert! :)
Pudding is completely different if we're referring to the AE use. It's a sweet made from cornflour, milk sugar etc and served at room temperature or refrigerated. Ice cream is of course frozen. "Pudding" is also a BE expression used for dessert in general. I think there would need to be a different wording here to accept this as a dessert. In other sentences, we use "dessert/sweet/pudding" interchanageably.
Trying to explain pronunciation can be tricky even in Greek where most letters are straight forward as they appear. It just happens the the letter "Γγ" is a bit tricky. So, to just go here where you can hear it pronounced by native Greek speakers. But no you would not pronounce this without the "g".