Yes, the pronounciation of "œuf" is really bad here. It seems she had hiccups. Let's report it with the "report" button! (Usually people are complaning about pronounciation, even if they are perfectly fine for natives, but here, you're right)
Note: There's an important thing to know about "œuf": you pronounce it "euf" (see my audio link) when it's singular, but the pronounciation change when it's "œufs" (plural), you would pronounce it "eu" without the "f" sound.
When you learn a word, make a point of learning the appropriate article. The article always makes the gender clear both in print and sound. Tie the article to the word so that it is a part of the imprint of the word that you form in your mind. If you think of la pomme not pomme you can't forget what gender it is. The article isn't an annoyance that has to be dealt with. It is your friend that enables you to easily remember gender.
There are no simple rules that make knowing the gender of new words easy to determine. You can improve your successful guess rate by remembering that there are approximately thirty per cent more masculine words than feminine in French, that many words that end with an e in French are feminine and that there is a historical general preference for the masculine usage in French.
Unfortunately what ever logic there was ages ago, when gender was being assigned to most words in French, is no longer visible to modern students.