Elephants that attack humans don't tend to live very long, especially during the film's time period. Since Mrs. Jumbo was considered a "mad elephant" worthy of chains and warning signs, it's likely they were keeping her locked away until she could be executed.
Then what is she doing in the back of the train at the end of the movie?
Simple; Dumbo is a star by then, and publicity is everything. They're not going to do anything bad to the famous flying elephant's mother, let alone kill her.
In Real Life, they probably wouldn't consider the relationship between Dumbo and his mom worth bothering to preserve, but then, this is a Disney movie, and this ending seems happier than the alternative.
Of course Dumbo, being able to fly, could've threatened them to let her go.
Timothy was clearly capable to communicate with humans, and at the end of the movie he was acting like Dumbo's representative and signing contracts for him. It is reasonable to assume one of the first pre-contractual conditions he demanded was Dumbo's mother was released and treated well.
They want to breed her back to Mr. Jumbo to have more flying elephants/more profit. If daddy's still alive that is - I always thought he was and they simply don't take a bull elephant with them on tour.
What'll happen to Dumbo when he grows too big for his ears to carry?
Maybe he'll merely be a retired Former Child Star, and perhaps be able to whip up strong winds with his ears instead?
It's possible that Dumbo's ears would grow in proportion to his body as he gets older. So his ears would still be massive compared to his body as an adult.
The square-cube law would mean that wouldn't really do much unless they grew REALLY big.
The square-cube law most certainly wouldn't allow for a flying elephant to begin with. I doubt it's in effect here.
Another relatively mild example. The skit Dumbo plays in with the clowns makes fun of a mother going hysterical because her child is in peril, while those around her are either indifferent or making things worse. It is an almost direct parody of the situation that led to Dumbo being separated from his mother in the first place. Not only has Dumbo lost his mother and been branded a clown, he is being forced to mock his own tragedy.
Imagine what would have happened if Dumbo hadn't been able to fly when he lost that feather. Yeah... let's just say it wouldn't have been pretty... (And for added horror, if he had missed the tarp below when they forced him out that one time. Sure, he might not have died, but injuries would be inevitable.)