For pet owners, the fact that Kyo hates leeks, onions, and miso makes perfect sense - those foods are dangerous for cats and should be never given to them. The Zodiac animal within Kyo is merely negatively reacting to what are, instinctively, health hazards.
You know that New Year's dance Yuki was so ashamed of? Well, it turns out that there are actually two members of the Zodiac that dance together, the "old year" and the "new year". Since the Ox follows the Rat, Yuki would have danced with Haru.
If you think about it, the real origin of God and the zodiac, the story of the zodiac that's known, and the state of the Sohma family makes a lot of sense. In the real story, the Cat was good-hearted, but God and the other animals didn't want to hear what it had to say. Instead, they chose to believe that the Cat was betraying them, and spent countless reincarnation cycles hating it. This is why the person possessed with the Cat is the scapegoat of the family - the rest of the Sohmas don't want to acknowledge all of the problems they have, so they vent all of their hatred on one person, for something that's not their fault. Meanwhile, the zodiac legend was told to show the Cat in as bad a light as possible - as a gullible idiot.
The Sohma Curse itself is one giant case of this. Once you realize that the actual curse isn't that they turn into animals, but that they are tied into an everlasting bond, where they both fear and love, hate and want, their relationship with their God and Family. It only gets better when the symbolism of this playing out like a Coming-of-Age Story is made clear. Everything about the Sohma Curse works to reinforce these themes in such a brilliant way, especially Kyo's place as The Cat as partially stated above, this troper can't ever stop thinking about it. "Time changes and people, too. And what was once a bond became a curse."
While Kyo could never defeat Yuki in a fight, Kyo did win against Yuki in the sense that he was the one Tohru fell for. Yuki even points out in-universe how he could only make Tohru react like a blushy little girl, while with Kyo, she acted more like a 'woman', and Yuki would try and fail to get the same response from her.
The effects of Akito mind raping Kana. Allegedly, after the ordeal, she spent all of her time sobbing and blaming herself for Hatori's eye injury, no matter what Hatori said to her. Don't get it yet? Kana was driven to clinical depression.
How can Yuki consistently best Kyo in fights, but be so weak and ill the rest of the time? You might possibly chalk it up to an adrenaline rush, but the backlash afterwards should put him in the hospital.
Unless this troper missed something, Yuki was only very sickly and weak when he was a child. By the time Tohru (and the audience) sees them fighting, and Yuki kicking Kyo's ass, they're both young adults, and Yuki seems very healthy and active.
As I recall, he still has weak spells (They have to carry him to school and bring Hatori in to examine him at one point.) Maybe he has improved, but it still seems a stretch that he could match a martial artist that's constantly training...
You don't have to be strong to be a skilled martial artists, it's more about technique. Yuki is alot more tranquil and focused than Kyo which usually makes for a better fighter than one who is driven by simple rage and just blindly attacks.
Also, the cat can never defeat the rat— when Kyo and Haru first fight, Yuki explicitly says to Tohru that Kyo probably is stronger against Haru. This seems to imply that the curse is interfering whenever Yuki and Kyo fight.