The sound is a bit weird to me, it's more like "hooit" rather than "hoort". Is it only me or are there other people having problem with this?
The 'r' in hoort is quite silent indeed, for instance if you compare hoort to the word 'raar' (weird). I think it may depend on the accent but overall it is pronounced silent in the Netherlands. I don't think that it sounds like Hooit though, if a dutch person would read 'hooit' then (s)he will prounce the 'I' like an 'J' or like the 'i' in 'Hoi' (Hi).
This is awkward in English. I would say "She doesn't hear anyone." Is this actually how this would be said in Dutch, or is this just Duo being goofy?
Yes, "zij hoort niemand" is how it would be said in Dutch. The English translation is indeed a little awkward here.
"She hears no one" This is also an acceptable translation and removes any awkwardness in my opinion.
The present tense for the verb "horen" is: ik hoor, jij hoort, hij/zij/het hoort, wij horen, jullie horen, zij horen. "Ze" can be either the 3rd person singular or 3rd person plural. But the conjugaison stem + t indicates that it is the 3rd person singular.