Translation:Do you remember this kanji?
If only you are wondering why you need to use the continuous form ~ています, it is because the concept of remember is different in Japanese from English. If we say something is in our memory, in English we say I remember it but in Japanese we need to say I am remembering it. 覚える is more like to memorize something and once you memorized it you are in the continuous state of remembering that thing.
It is formal.
おぼえているか？ is also viable.
With the context it makes more sense for it to be a formal sentence. Like a teacher will be asking a student if they remember the kanji which is a more formal situation.
You almost never use "ka" with the plain form. End it with "no", or no article at all, just a rise in the intonation. Using ka with the plain form is a very crude way of speaking, and shouldn't be used.
Can't help but thinking the meaning actually is: "Do you remember this feeling?"
I know there are two schools of thought on pluralizing Japanese loanwords and that neither can be proven correct over the other, but I am firmly in the camp of do not add an -s to kanji ever please thank you sorry..........
おぼえる can sometimes be translated as "learn" (or more often "memorize"), but おぼえている is better translated as "remember".
Kanji is the object of the verb, indicated by を. If there was a subject of the sentence, that would be indicated by は. For example, あなたはこの漢字を覚えていますか？ would be "do you remember this kanji?", while かれらはこの漢字を覚えていますか？ would be "do they remember this kanji?"