Agreed. It is "(Do) you have a brush?". Very common.
I'm from the South, so this is even more "proper" than I would speak to my friends casually. I easily could further truncate this to simply "Got a brush?" and be totally within my rights. Not sure if them Yanks talk like that though. What I do know, is both of the aforementioned phrases are correct within my vernacular.
The "have" in "have you <verb>" is the "present perfect", nothing to do with possession (which you replaced by your "got", anyhow).
The Spanish sentence is a simple present tense and thus your translation does not match. Similarly, you couldn't say "you had", or "you will have" or "you will have had", etc. (Those are simply more obvious.)
They're both absolutely complete and correct. We use interrogatives that omit the auxiliary "do" quite often. However, there is a slightly different meaning to these questions. We use this structure to convey surprise or disbelief. We use the auxiliary with questions introducing new topics.
Cepillo is no ordinary brush. You would call a paint brush brocha or pincel. Cepillo, most of the time, will be a hairbrush or a cleaning brush. If you want refer to a toothbrush out of context you should say el cepillo de dientes. Because of that I think both brush or hairbrush should be acceptable translations.
Oh gosh, here we go again. One person gets a typo or Duolingo glitches and messes up, and, instead of just telling that ONE person to report it, we need a WHOLE DISCUSSION on slang, Italian and toothbrushes. PLEASE can we just say those 2 words, "Report it." instead of a full blown argument. Also, guys it's not cool to reeat what other people said 5 comments ago, hoping no one will notice, to look smart. Come on.
Content here does not indicate if the brush is for general purpose, personal grooming or house cleaning. The previous statement said "He is cleaning the oven." Content in this context could indicate scrubbing brush. Other questions were about razors, so hair brush could be useful. Go figure.
Tienes must be used with the pronoun "tú." You could use the conjugation tiene if you were translating this sentence with the formal "usted," rather than "tú" and it would also be correct. However, I think there are probably limited instances in real life where you would ask such a question in a formal setting (though practicality of sentence is not always Duo's game, so that doesn't really matter here.)
So for this type of question, you have to say "Do you have a brush?". If Duolingo wants you to not use casual conversation terms so you can't say "You have a brush", then why is "Have you got a brush?" incorrect? Also, I don't think that "You have a brush?" should be marked as wrong. If you were talking to someone above you in station, then you wouldn't say, "Do you have a brush?" because that we be weird to say to someone. You would only use it for someone equal or inferior to you in rank, so why couldn't you use the bit more casual term?