They're all fine, as is simply restating without comment:
He’s going to school— she’s going to school at Princeton.
On this site it may be worth observing that nobody expects spoken English to be perfectly expressed, from native speakers or anybody else. Speech is improvised: people start a sentence with only a general idea of where it’s going, change their minds in the middle, have second thoughts, find new ideas or expressions suddenly occurring. So if you listen closely to conversations, or if you read a literal transcript, you will find that errors are abundant. On the page, excited speech may be almost incomprehensible.
Consequently, most errors in speech are ignored, by both speakers and hearers. It’s taken for granted that errors will occur, and for the most part hearers make the necessary corrections automatically, from context.
Back when I was active in theatre I always told my actors to play through errors as if they hadn’t happened—“The audience’ll never know you screwed up unless you tell them.” The same principle operates in ordinary conversation: don’t make a correction at all unless there’s a real danger that what you said will be misunderstood. In your example, for instance, there's no need to back up unless two people, one male and one female, are under discussion.
I would expect an answer, although I wouldn't expect more than a polite "Fine, thanks" or similar.
I think most English speakers would answer with something, but the question is usually out of politeness and a full answer isn't usually expected. Unless you're particularly close, they probably wouldn't give a long-winded explanation, in the same way you first answered “I am well, thank you. And you?”
If they don't answer, I wouldn't really worry about it as it's usually a courtesy to the other person, and I'd just continue the conversation as normal.
An exception: My mum once told me she went to visit the doctor. He asked: "Hello, how are you?" She answered: "Fine, thanks." Then after a moment realised it wasn't just a polite question and said: "Oh, are you really asking? Well, I... [the reason for visiting the doctor]".
Best Answer
Here are the options that I can think of:
I will call the other person Bob.
To reply hoping the Covid-19 pandemic will be better:
Option 1:
^ above is hoping that we can start traveling again next year.
Option 2:
^ above is hoping that the vaccine will works and help fading the pandemic.
Option 3:
^ above is hoping that we can all get out of the lock down and start working again.
Option 4:
^ the above is just hoping that the whole pandemic will end.
To reply hoping the whole year would be better:
Option 1:
^ above is just mentioning that the next year can't be any worse than this year.
Option 2:
^ above is hoping that the next year would be completely different and better than this year.
Option 3:
^ above is saying that we need to make the next year a really good year.
Option 4:
^ above is saying that we can do it this year, make it meaning that we will make this year good.
There are my suggestions of replying to that sentence.