Not in a romantical way, in world of anime. They never made any association in manga, only in Bleach fillers, on which Toshiro appears to care for Karin's sake and is friends to some point.
They hold no romantic feelings towards each other, and this couple in my opinion only appeared because of Yuzu's assumption/fantasy of them getting married when they grow up.
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I think it's episode 132, because it's called "Hitsugaya, Karin, and The Soccer Ball".
No. This event has not happened.
Apperently, Hitsugaya did not die. He was only wounded and as you watch and read the series, you see that he is still alive. toushiro hitsugaya, in his human life, died by a plain of ice. he was then reborn in sole socity (hope i spelt that right) and later on, became a shinigami.
None episode.
No one knows, but he does have a grandma though.