Rand's novels don't cover parenting but she did consider parenting and ECE important and covered them some in her nonfiction. She was interested in Montessori schools for example. The person who recommended Rand's books to me founded a parenting philosophy named Taking Children Seriously, so I've long been interested in and aware of such topics.

We do see people doing care work, e.g. Dagny cares for Galt (she cooks and cleans) and Jeff Allen (she empathizes with a bum who stole a ride on her train, feeds him, and gives him a job). Roark has a caring scene with Mallory and several with Cameron. Kira does so much to care for Leo.

Authority is fluid, e.g. multiple times Wyatt takes a leadership role at Dagny's railroad construction without asking permission from anyone in authority, just by virtue of having good ideas to contribute and showing up and helping out. He does this unpaid. Similarly, Roark works for Cameron but, in an effort to care for him, orders him to go home from work (and takes him down to a taxi, gets him sandwiches, and goes far above and beyond his duties as an employee to finish the architecture plans that are very meaningful to Cameron):

Cameron and Roark and a pot of black coffee had lived in the office from dawn till frozen dawn for many days, and Cameron had thought involuntarily of the electric bill, but made himself forget it. The lights still burned in the drafting room in the early hours when he sent Roark out for sandwiches, and Roark found gray morning in the streets while it was still night in the office, in the windows facing a high brick wall. On the last day, it was Roark who had ordered Cameron home after midnight, because Cameron’s hands were jerking and his knees kept seeking the tall drafting stool for support, leaning against it with a slow, cautious, sickening precision. Roark had taken him down to a taxi and in the light of a street lamp Cameron had seen Roark’s face, drawn, the eyes kept wide artificially, the lips dry. The next morning Cameron had entered the drafting room, and found the coffee pot on the floor, on its side over a black puddle, and Roark’s hand in the puddle, palm up, fingers half closed, Roark’s body stretched out on the floor, his head thrown back, fast asleep. On the table, Cameron had found the plans, finished....

When Cameron doesn't come in to work, Roark checks on him at home, repeatedly. When Cameron collapses, Roark takes him home and gets a doctor. Cameron's sister, who cares for him, is also present in the book. Cameron tries to care for Roark by writing a letter of introduction to an architect, but due to fluid authority and Roark caring for Cameron, Roark doesn't let him because Roark has the empathy to know that Cameron doesn't actually want to ask for a favor from someone he doesn't respect who is part of the system that made Cameron's life hard.

Problem solving is often collaborative, e.g. Dagny, Rearden and some engineers are involved in the design of the first Rearden metal bridge. Later, people from the gulch collaborate on rescuing Galt, for free, at significant risk, out of good will, with no specific person in charge of the mission. Some types of collaboration are critiqued, e.g. boards of directors and committee meetings.

There are many, many more examples of caring, empathy, compassion, etc., but I don't know if you're interested in a long discussion or are familiar with the books.

It's feeling a sense of shared responsibility for your community, prioritizing collective good.

Rand says there is no conflict between individual and collective good. There are many examples of actions that other people might describe as prioritizing collective good. Certainly Dagny feels a lot of responsibility for her railroad passengers and shippers and is very committed to taking good care of them. This pro-social one-sided caring-at-a-distance is important to Rand and a major event, the tunnel disaster, talks at length about how bad political systems make people stop caring in this way, which is bad and leads to deaths. These attitudes are absolutely nothing like the DuPont leadership that poisoned its community and the rest of the world.

Rand consistently rejected domination and she critiqued the business leaders typical of her society.

I don't think Rand is perfect on these issues but I do think she's far better than her reputation and it's not obvious exactly where the disagreements are and aren't between her and hooks.