Ich brauche is quite a direct thing to say. In a restaurant it would be interpreted with some sense of urgency, like after you just ate something very hot and immediately wanted to cool your mouth down.
You will very commonly hear Ich möchte [ein Glas Wasser] which means "I would like [a glass of water]". You can just learn this as a verb directly if you want, but technically it is the Subjunctive II form of Ich mag ("I like"). Polite requests use verbs in this modified way. That's a bit advanced and Duolingo teaches it a bit later.
Yes, but usually you would have to have some reason for saying it that way, as it puts extra emphasis on "water". Like, if you were rescued from the desert and reached for a bottle to take a drink only to discover that it's chilli sauce. WASSER brauche ich! Chilisoße nicht!, you might say. It's like saying "It is water that I need" in English.
No, "brauchen" does not necessarily imply urgency. If you cook a recipe, you may say "Und jetzt brauche ich 250 ml Wasser". = "And now I need 250 ml of water". It is just what the situation (or recipe) is asking for.
Loosely translated, "I want some water" = "Ich will etwas Wasser (haben)" can be interpreted as someone needing or wanting water. We try to keep translations natural, yet so that they still match closely the source sentence.
It sounded like "prauche" which was then marked correct. I came here and see it should be "brauche".
It would help if we could be told about German spelling errors like we are with English spelling errors as I had the same problem putting an "h" into "lehrnen" instead of "lernen".