No, but an infant (alien, looks like a plush candle flame with four legs and three eyes) is separated from its parents, and the parents THINK it's been abducted.
Buck Cluck didn't care about his son until he won the baseball game, and then goes back to shunning him the second time that Chicken Little didn't prove that the sky was falling. Runt's mother is also pretty mean to him.
It's evident that the Chicken Little's mother died before the movie's events. His dad talks about wishing that she were still there and looks sadly at photos of her. We're not told when or how she died.
There's the old "person is too fat to fit through a hole and gets helplessly stuck, and others struggle to push/pull them through" thing. In this case, as in so many others, the person being stuck puts others in danger, since they all need to get away from the bad guys fast, and the hole is the only exit. So the fat person is portrayed not only as pathetic and useless, but also as a burden and a liability. It made me angry.
For the most part, no. But one (albeit mean) side character has her personality permanently altered and the other characters treat this as a good thing. I'll let you decide what to make of that.