I'm not going to post them all at once, so the top three go first!
Her father died of an undefined illness. He got sick.
Her "first love" was an unknown boy who helped her find her house when she was lost, alone and upset. He was wearing a red cap and gave it to her before he left. She still has the cap and believes that she will meet him again. Hana-chan and Uo-chan know of the boy and Kyou overheard her talk about him.
Hiro once asks why she carries her mother's picture around. She goes serious, but doesn't answer, switching back to normal immediately.
As of now, her relationship with Kyou is better in terms of closeness and friendship, but Yuki seems to charm her more.
She admits to herself that it's getting harder and harder for her heart to resist them (yes, plural).
He was the "red cap" boy. He knows he was, but he hasn't told Tohru yet, and no one else knows.
He still calls her Honda-san.
His parents and brother "sold him out" instead of going against Akito.
His mother called him a "tool" and would beat him.
When Akito locked him up, he waited and waited for his parents and/or brother to save him. They never did. How he got out is undetermined. Likely Akito just got bored.
After the lock-up incident, he stopped talking for a time (like Kisa). Unsure how he overcame this.
Akito is giving him until the end of high school, then he will be forced to 'return home'.
Akito "needs" him and says that Yuki should be grateful for how much he "needs" him.
There's something in Yuki's mind/memory that he's locked away. While what it is is unrelieved, the lid is openeding in the manga, thanks to an off-screen scene with Akito and Kureno. Afterwards, Yuki is a lot more confident and forward with Tohru (even kissed her on the forehead and hinted that he would "tell her" soon). He also told Kyou not to do or say anything with Akito that would endanger Tohru. Shigure asked what Akito told him and all he answered was "The truth."
Hatsuharu sees him soon after the incident and comments on the change in him, then says, "Leaving the nest?" And Yuki smiles and comments how Haru knows him better than anyone.
Rin calls him "Akito's Toy."
His vengeful cat spirit form does vanish after time, independant of rain or presense of talisman. Those are just factors that can trigger the transformation. It will fade even if the rain doesn't stop or if the bracelet isn't returned.
His bracelet is a talisman made from human bones and dyed in blood. It has been passed down by the cat for generations.
His father is still alive and hates him. He blames Kyou for the death of his wife.
Kyou's mother commited suicide.
It is hinted that Kyou's mother is in some way connected to Kyoko, Tohru's mother, but it is unclear how close the connection is at this point, though it is assumed that Kyou saw a picture of Tohru once before his mother died.
His father is in contact with Kazuma and is working towards the future imprisionment of Kyou.
Kyou has a bet with Akito. If he defeats Yuki before the end of the school year he will be recognized as a Jyunishi. If he doesn't, he will be locked in a cage for the rest of his life. Kazuma knows of this bet, and Tohru as well now (Akito revealed it to her). It is unclear if Yuki knows, but it seems likely. Shigure and Hatori probably know as well.
Kyou does admit (to himself) that he is in love with Tohru. He happens to do this in front of Akito, but he denies it when Akito questions. To protect her, he decides it is better to lock his feelings away.