"Do you remember?"
"Anata" can be rude. They really shouldn't have that for a sentence like this.
Yes, it can be rude, and I have heard this from several native speakers. Second person pronouns are generally seen as too direct in Japanese.
The most common usage I see of あなた is by wives to their husbands. The average person would be uncomfortable with that level of intimacy, so I would consider that rude.
That's because it wants you to use the ている form for "the state of remembering," rather than the non-past ます form ("will remember/remember")
It is not correct because you're supposed to use the te-iru form of the verb. See Ever2662's comment above
For those of us who duolingo decided not to give the audio for the kanji "覚えて" is "おぼえて".