You got the cases right in all three sentences.
I'll try to provide English translations which would be as close to the literal meaning of the Russian phrases as possible.
Please note that they are not actual translations, they are English approximations of grammatical structure of the original Russian phrases.
У меня́ есть де́ньги. // There are money next to me.
Russian uses proximal possession: Russians don't say "I have something", they say "there is something next to me" to convey the same meaning.
In English, you would have said "I have him" but "he's next to me".
If you look closely you can see that you use "him" in the first sentence but "he" in the second one.
This is because grammatically, "him" is an object in the first case, but a subject in the second case. Similar logic applies to Russian, except that in Russian almost all nouns decline, not just personal pronouns.
У меня́ доста́точно де́нег. // I have enough of money.
In English you would normally use "I have enough money", but there are cases when you are actually using "of": "I've heard enough of this nonsense".
If we go back to the personal pronouns analogy, you would have said "I have enough of him" in English. "Of something" mostly (not always, but mostly) translates to Russian genitive, and that's why you're seeing it here.
Вы мо́жете дать мне немно́го де́нег? // Can you give me a little of money?
Same logic as above here.
But I expect it to be in the nominative because "give something" and the something would be in nominative
In Russian, a direct object to дать ("to give"), or to any other verb for that matter, is never nominative. However, it's not the reason денег is put into genitive here. The real reason is that it's not денег that is the direct object, but немного.
Денег complements the немного and it's the latter which governs the case of the former. And немного requires a genitive.