"Do you think most robins bounce their babies a little when they fly off the nest? Can this hurt the eggs or nestlings?"
Most robins actually bounce their babies MORE than these robins do, because most robin nests are on branches of trees, which bounce a lot more than a ladder does! This has been happening for so many thousands of years that baby robins are designed to live with it.

"What is the mother feeding these tiny nestling?"
The mother is regurgitating small bits of the food she has caught and eaten herself. They are so tiny that a whole worm would be too big for their mouths and throats, and their stomachs aren't ready to digest raw food yet. So her stomach gets the process started. This is also how their stomach and intestines will get the right bacteria in them so as they get bigger they can feed themselves and digest all their own meals.

"What is the mother eating in the second and third video clips?"
The mother is eating the babies' fecal sacs--their poop! The babies are so tiny that they aren't digesting all their food, and their intestines don't have much bacteria yet, so this helps the mother to have enough food while she's spending most of her energy getting food for the babies.

"Why does the female open her beak when the male flies in?"
The female is a little stressed when ANYTHING approaches the nest when the babies are so tiny. So her open beak may show a little fear and a little warning. But it also helps her to beg from him. Her hormones are all geared up to nurture babies. If the male feeds her, she can use that food to feed the babies.

"What food is the male bringing?"
It's hard to see it for sure. He has definitely swallows worms and or bugs, because he has to wiggle his throat to make them come up again. It looks like he may have given her a small worm or just the stringy, goopy remains of the food he's eaten.

"Why doesn't the male feed the babies?"
The male is perfectly capable of feeding them. But at this point, the mother's hormones are at such a high level that she is more protective of and anxious to feed the babies than he is, so she is the one who does that job.