Did Hiei have a necklace his mother gave him?

Yes, there are certified pictures drawn by Yoshihiro Togashi(the creator of Yu Yu Hakusho) showing Hiei with the necklace made of his mother's tears.

Hiei can't catch human illnesses, right?

Correct. Hiei is a youkai. Ningen illness shouldn't be a problem.

Is Hiei 1/2 Koorime?

Although Hiei's mother was indeed Koorime, Hiei is not. If you did your homework, you would know that "Koorime" translates to "Ice Maiden". Coincidently the ice country Koorime, was all females. And last time I checked, Hiei was male, so how can he be an ice maiden?

Can Hiei cry gems, like his sister?

No. Only Koorime can cry gems. And if you read the previous question, Hiei isn't Koorime.

Was Hiei born with his Jagan eye?

Someone actually asked me this...No, He got it implanted by a demon named Shigure(See Hiei's Biography)

Are Hiei and Kurama a couple?

Well, to tell you the truth, no one knows. It seems like they could be, but then again, they may be really close friends. Yoshihiro Togashi(the creator of Yu Yu Hakusho) said that when he made them, he didn't intend for them to be a couple, but he did like the idea of it. So the battle of anti Hiei x Kurama-ists, and pro Hiei x Kurama-ists shall continue.

Are Hiei and Mukuro a couple?

Hopefully not. Many people assumed they are, because she hugged him once, but Hiei couldn't fight back. There really aren't any facts to base a true/false answer for this.